Fix a matrix A and a vector b. Suppose that y is any solution of the homogeneous system Ax=0 and that z is any solution of the system Ax=b. Show that y+z is also a solution of the system Ax=b.
Answer:
Since y is a solution of the homogeneus system then satisfies Ay=0.
Since z is a solution of the system Ax=b then satisfies Az=b.
Now, we will show that A(y+z)=b.
Observe that A(y+z)=Ay+Az by properties of the product of matrices.
By hypotesis Ay=0 and Az=b.
Then A(y+z)=Ay+Az=0+b=b.
Then A(y+z)=b, this show that y+z is a solution of the system Ax=b.
Ms. Jones deposited $100 at the end of each month for 10 years into a savings account earning 6% interest compounded monthly. However, she deposited an additional $1000 at the end of the first year. How much money was in the account at the end of the tenth year?
Answer:
$1895.64
Step-by-step explanation:
Given:
Principle for the first year = $100
rate of interest = 6% compounded monthly
thus,
rate per month, r = \frac{\etxtup{6}}{\textup{12}}= 0.5% = 0.005
Total time = 10 year\
Now,
for the first year
number of months, n = 12
Amount at the end of first year = Principle × ( 1 + r )ⁿ
on substituting the values, we get
Amount at the end of first year = 100 × ( 1 + 0.005 )¹²
or
Amount at the end of first year = $106.17
Therefore,
The principle amount for the consecutive years will be
= $1000 + Amount at the end of first year
= $1000 + $106.17 = $1106.17
Thus, for the rest 9 years
n = 9 × 12 = 108
Principle = $1106.17
Final amount after the end of 10th year = Principle × ( 1 + r )ⁿ
or
Final amount after the end of 10th year = $1106.17 × ( 1 + 0.005 )¹⁰⁸
or
Final amount after the end of 10th year = $1895.64
If $5000 is deposited into an account which pays 6% compounded
monthly, approximately how much money will be in the account at the
end of 8 years?
Answer:
The amount in the account at the end of 8 years is about $8070.71.
Step-by-step explanation:
Given information:
Principal = $5000
Interest rate = 6% = 0.06 compounded monthly
Time = 8 years
The formula for amount after compound interest is
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
where,
P is principal.
r is rate of interest.
n is number of times interest compounded in a year.
t is time in years.
Substitute P=5000, r=0.06, n=12 and t=8 in the above formula.
[tex]A=5000(1+\frac{0.06}{12})^{(12)(8)}[/tex]
[tex]A=5000(1.005)^{96}[/tex]
[tex]A=5000(1.61414270846)[/tex]
[tex]A=8070.7135423[/tex]
[tex]A\approx 8070.71[/tex]
Therefore the amount in the account at the end of 8 years is about $8070.71.
An airplane over the Pacific sights an atoll at 20 degree angle of depression. If the plane is 425 m above water, how many kilometers is it from a point 425 m above the center of the atoll?
Answer:
[tex]1.167 km[/tex]
Step-by-step explanation:
We are given with-
Height of airplane from water [tex]a[/tex] = [tex]425 m[/tex]
Angle of depression (∅)= 20°
Now,
[tex]tan(20) = \frac{b}{a}[/tex]
[tex]a = \frac{b}{tan(20)} \\a = \frac{425}{tan(20} \\a = 1167.677 m[/tex]
[tex]a = 1.167 km[/tex]
Suppose that scores on a test are normally distributed with a mean of 80 and a standard deviation of 8. Answer the questions below. (a) What is the 70th percentile? (round to the tenths place) (b) What percentage of students score less than 70? (round to the tenths place, give the percent)
Answer:
(a) 84.2
(b) 10.6
Step-by-step explanation:
To solve this questions we can use the standardization formula, where we know that if [tex]X\sim N(\mu,\sigma^2)[/tex] then [tex]Z=\frac{X-\mu}{\sigma} \sim N(0,1)[/tex]
So for
(a) we know that the z score for the 70th percentile is 0.524, so using the normalization equation we have
[tex]\frac{X-\mu}{\sigma}=0.524[/tex]
[tex]X=0.524*8+80=84.192[/tex]
(b) We can procede as above and get
[tex]P(X<70)=P(\frac{X-80}{8}<\frac{70-80}{8})=P(Z<-1.25)=0.1056[/tex]
Find an equation of the line L that passes through the point (-8, 4) and satisfies the given condition. The x-intercept of L is -10.
Final answer:
To find the equation of a line that passes through a given point and has a given x-intercept, we can use the point-slope form of a line.
Explanation:
To find the equation of a line that passes through the point (-8, 4) and has an x-intercept of -10, we can use the slope-intercept form of a line, which is y = mx + b.
First, let's find the slope of the line using the given information. The x-intercept represents the point where the line crosses the x-axis, so if the x-intercept is -10, we know that the point (-10, 0) is on the line.
Using the formula for slope, which is m = (y2 - y1) / (x2 - x1), we can calculate the slope of the line as (0 - 4) / (-10 - (-8)) = -4 / -2 = 2.
Now, we can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line.
Substituting the values (-8, 4) and m = 2 into the equation, we have y - 4 = 2(x - (-8)).
Simplifying the equation, we get y - 4 = 2x + 16.
Finally, isolating y, we arrive at the equation of the line: y = 2x + 20.
If we assume the population of Grand Rapids is growing at a rate of approximately 4% per decade, we can model the population function with the formula
P( t ) = 181843 ( 1.04 )^(t / 10).
Use this formula to compute the average rate of change of the population on the intervals [ 5 , 10 ] , [ 5 , 9 ] , [ 5 , 8 ] , [ 5 , 7 ] , and [ 5 , 6 ] .
Answer:
The average rate of change of the population on the intervals [ 5 , 10 ] , [ 5 , 9 ] , [ 5 , 8 ] , [ 5 , 7 ] , and [ 5 , 6 ] are 734.504, 733.06, 731.62, 730.185 and 728.75 respectively.
Step-by-step explanation:
The given function is
[tex]P(t)=181843(1.04)^{(\frac{t}{10})}[/tex]
where, P(t) is population after t years.
At t=5,
[tex]P(5)=181843(1.04)^{(\frac{5}{10})}=185444.20[/tex]
At t=6,
[tex]P(6)=181843(1.04)^{(\frac{6}{10})}=186172.95[/tex]
At t=7,
[tex]P(7)=181843(1.04)^{(\frac{7}{10})}=186904.57[/tex]
At t=8,
[tex]P(8)=181843(1.04)^{(\frac{8}{10})}=187639.06[/tex]
At t=9,
[tex]P(9)=181843(1.04)^{(\frac{9}{10})}=188376.44[/tex]
At t=10,
[tex]P(10)=181843(1.04)^{(\frac{10}{10})}=189116.72[/tex]
The rate of change of P(t) on the interval [tex][x_1,x_2][/tex] is
[tex]m=\frac{P(x_2)-P(x_1)}{x_2-x_1}[/tex]
Using the above formula, the average rate of change of the population on the intervals [ 5 , 10 ] is
[tex]m=\frac{P(10)-P(5)}{10-5}=\frac{189116.72-185444.20}{5}=734.504[/tex]
The average rate of change of the population on the intervals [ 5 , 9 ] is
[tex]m=\frac{P(9)-P(5)}{9-5}=\frac{188376.44-185444.20}{4}=733.06[/tex]
The average rate of change of the population on the intervals [ 5 , 8 ] is
[tex]m=\frac{P(8)-P(5)}{8-5}=\frac{187639.06-185444.20}{3}=731.62[/tex]
The average rate of change of the population on the intervals [ 5 , 7 ] is
[tex]m=\frac{P(7)-P(5)}{7-5}=\frac{186904.57-185444.20}{2}=730.185[/tex]
The average rate of change of the population on the intervals [ 5 , 6 ] is
[tex]m=\frac{P(6)-P(5)}{6-5}=\frac{186172.95-185444.20}{1}=728.75[/tex]
Therefore the average rate of change of the population on the intervals [ 5 , 10 ] , [ 5 , 9 ] , [ 5 , 8 ] , [ 5 , 7 ] , and [ 5 , 6 ] are 734.504, 733.06, 731.62, 730.185 and 728.75 respectively.
A simplified model of the human blood-type system has four blood types: A, B, AB, and O. There are two antigens, anti-A and anti-B, that react with a person’s blood in different ways depending on the blood type. AntiA reacts with blood types A and AB, but not with B and O. Anti-B reacts with blood types B and AB, but not with A and O. Suppose that a person’s blood is sampled and tested with the two antigens. Let A be the event that the blood reacts with anti-A, and let B be the event that it reacts with anti-B. Classify the person’s blood type using the events A, B, and their complements.
Answer: Blood type will be A when event "A" happened and event "B" did not happen. Blood type will be B when event "A" did not happened and event "B" happened. Blood type will be AB when both events happened and blood type will be O when both events did not happen.
Step-by-step explanation:
S={AntiA reacts; AntiA does not react; AntiB reacts; AntiB does not react}
If AntiA reacts and AntiB reacts = AB (A∩B)
If AntiA does not react and AntiB does not react= O (A'∩B')
If AntiA reacts and AntiB does not react= A (A∩B')
If AntiA does not react and AntiB reacts= B (A'∩B)
The blood type is determined by observing the blood reaction to anti-A and anti-B antigens. Type A reacts with anti-A, Type B reacts with anti-B, Type AB reacts with both, and Type O doesn't react with either.
Explanation:The process of identifying a person's blood type using anti-A and anti-B antigens is straightforward. If the person's blood agglutinates (clumps together) when anti-A antigens are added, it means the blood has type A glycoproteins on the surface and the person has type A or AB blood. This is what we call event A.
Similarly, if the blood reacts with the anti-B antigen (event B), it means the person has type B or AB blood. If the blood reacts to both anti-A and anti-B antigens, it must be type AB. If the blood doesn't react with either antigen (the complement of both A and B events), it signifies the person has type O blood, which lacks both A and B glycoproteins on the erythrocyte surfaces.
It's also worth noting that AB blood can accept blood from any type (universal acceptor), while O blood type can be transferred to any blood type (universal donor) as it doesn't cause an immune response due to lack of A and B antigens.
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When entering large numbers in the answer box, do not use commas. For example, enter 1276400 for the number 1,276,400. Do not enter 1,276,400. If you accidentally enter commas, you will receive feedback as a reminder. Answer the following question by typing the numeric answer into the answer box. What is the sum of 9260 and 3240?
Answer:
The answer is 12500....
Step-by-step explanation:
We have been asked that what is the sum of 9260 and 3240?
The sum of two numbers is the result you obtain by adding the two numbers together.
Addition is the mathematical process of putting things together. The plus sign "+" shows that numbers are added together. We start adding the numbers from right hand side.
We have two values 9260 and 3240. We will add these two values together.
9 2 6 0
+ 3 2 4 0
__________
12 5 0 0
Thus the answer is 12500....
An experimenter is studying the effects of temperture, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration.
(a) If any particular experiental run involves the use of single temperature, pressure, and catalyst, how many experimental runs are possible?
(b) How many experimental runs are there that involve use of the lowest temperature and two lowest pressure?
Answer:
Step-by-step explanation:
Given that an experimenter is studying the effects of temperture, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration.
a) Experimental runs possible if use of single temperature, pressure and catalyst is there = no of temperatures x no of pressures x no of catalysts
= [tex]3*4*5 = 60[/tex]
b) Here pressure and temperature have no choice as lowest is selected.
no of methods = no of catalysts x 1 x1
= 5
Problem 2.191 A researcher receives 93 containers of oxygen. Of those containers, 20 have oxygen that is not ionized and the rest are ionized. Two samples are randomly selected, without replacement, from the lot. Round your answers to three decimal places (e.g. 98.765). (a) What is the probability that the first one selected is not ionized? (b) What is the probability that the second one selected is not ionized given that the first one was ionized? (c) How does the answer in part (b) change if samples selected were replaced prior to the next selection? Find the probability. (d) What is the probability that both are ionized?
Answer:
Step-by-step explanation:
Given that there are 20 non ionized containers and 73 ionized containers
Two samples are drawn without replacement
a) the probability that the first one selected is not ionized=[tex]\frac{20}{93} =0.215[/tex]
b) the probability that the second one selected is not ionized given that the first one was ionized
= When first one was ionized we got left over as 20 and 72
Hence = [tex]\frac{20}{92} =0.217[/tex]
c) If with replacement left over 20 and 73 and hence prob = 0.215 as in part a
d) the probability that both are ionized=[tex]\frac{73C2}{93C2} =0.614[/tex]
An IV fluid contains 75 mg of Drug B in 500 ml of the IV fluid. If the patient needs 300 mg of this medication, how much of this IV fluid should be infused
Answer:
2000ml = 2L of this IV fluid should be infused.
Step-by-step explanation:
This problem can be solved by a simple rule of three, in which the relationship between the measures is direct, which means that there is a cross multiplication.
The problem states that each 75 mg of the medication contains 500 ml of IV fluid. How many ml of IV fluid are there in 300 mg of the medication?
So
75mg - 500ml
300 mg - x ml
[tex]75x = 300*500[/tex]
[tex]x = \frac{300*500}{75}[/tex]
[tex]x = 4*500[/tex]
[tex]x = 2000[/tex]ml
2000ml = 2L of this IV fluid should be infused.
To administer 300 mg of Drug B, given that 75 mg is in 500 ml of IV fluid, you need infuse 2000 ml of the IV fluid. This was calculated using proportional relationships based on the concentration of the drug in the fluid.
To determine how much IV fluid should be infused to provide the patient with 300 mg of Drug B, we can use a simple proportion based on the concentration of the drug in the IV fluid.
We know that 75 mg of Drug B is contained in 500 ml of the IV fluid.This corresponds to a concentration of 75 mg/500 ml or 0.15 mg/ml.To find out how many milliliters (ml) are needed for 300 mg, we set up the following proportion:0.15 mg/ml = 300 mg / X ml
Solving for X, we get:
X = 300 mg / 0.15 mg/ml
X = 2000 ml
Thus, 2000 ml of the IV fluid should be infused to provide the patient with 300 mg of Drug B.
Explain why the formula is not valid for matrices. Illustrate your argument with examples. (A + B)(A − B) = A2 − B2 The formula is not valid because in general, the distributive property is not valid for matrices. The formula is not valid because in general, B(−B) ≠ −B2 for matrices. The formula is not valid because in general, AB ≠ BA for matrices. The formula is not valid because in general, A(−B) ≠ −AB for matrices. Select the pair of matrices, A and B, for which the formula is not valid.
Answer:
The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices.
Step-by-step explanation:
The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices. That is, AB does not necessarily equal BA.
[tex](A+B)(A-B) = A^2-AB+BA-B^2\neq A^2 - B^2[/tex]
[tex]A=\left[\begin{array}{ccc}1&0&0\\0&0&6\\0&8&0\end{array}\right] \\B=\left[\begin{array}{ccc}0&2&0\\6&0&0\\0&0&9\end{array}\right] \\(A -B) = \left[\begin{array}{ccc}1&-2&0\\-6&0&6\\0&8&-9\end{array}\right]\\\\(A + B) = \left[\begin{array}{ccc}1&2&0\\6&0&6\\0&8&9\end{array}\right]\\(A - B)(A + B) = \left[\begin{array}{ccc}-11&2&-12\\-6&36&54\\48&-72&-33\end{array}\right]\\A^2 - B^2 = \left[\begin{array}{ccc}-11&0&0\\0&36&0\\0&0&-33\end{array}\right]\\[/tex]
You can use the fact that multiplication of matrices is dependent on the order of the matrices which are multiplied.
The correct option for the given condition is
Option C: The formula is not valid because in general, AB ≠ BA for matrices.
Why is it that AB ≠ BA for two matrices A and B usually?It might be that AB = BA for two matrices A and B but it is very rare and thus, cannot be generalized as identity.
Suppose A has got shape (m,n) (m rows, n columns)
and B has got shape (n,k) (n rows, k columns), then AB is defined but BA is not defined if k ≠ m.
Also, even if k =m, we can't say for sure that AB = BA
Thus, usually we have AB ≠ BA
Using the above fact to and distributive property to evaluate (A + B)(A − B)For two matrices A and B, supposing that AB and BA are defined, then we have
[tex](A+B)(A-B) = A(A-B) + B(A -B) = A^2 -AB + BA - B^2[/tex]
Since may or may not have AB equal to BA, thus, we cannot cancel those two middle terms to make 0 matrix.
Thus,
The correct option for the given condition is
Option C: The formula is not valid because in general, AB ≠ BA for matrices.
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Find the projection of the vector A = î - 2ġ + k on the vector B = 4 i - 4ſ + 7k. 15. Given the vectors A = 2 i +3 ſ +6k and B = i +59 +3k. How much of vector B is along vector A?
Answer:
Part 1)
Projection of vector A on vector B equals 19 units
Part 2)
Projection of vector B' on vector A' equals 35 units
Step-by-step explanation:
For 2 vectors A and B the projection of A on B is given by the vector dot product of vector A and B
Given
[tex]\overrightarrow{v_{a}}=\widehat{i}-2\widehat{j}+\widehat{k}[/tex]
Similarly vector B is written as
[tex]\overrightarrow{v_{b}}=4\widehat{i}-4\widehat{j}+7\widehat{k}[/tex]
Thus the vector dot product of the 2 vectors is obtained as
[tex]\overrightarrow{v_{a}}\cdot \overrightarrow{v_{b}}=(\widehat{i}-2\widehat{j}+\widehat{k})\cdot (4\widehat{i}-4\widehat{j}+7\widehat{k})\\\\\overrightarrow{v_{a}}\cdot \overrightarrow{v_{b}}=1\cdot 4+2\cdot 4+1\cdot 7=19[/tex]
Part 2)
Given vector A' as
[tex]\overrightarrow{v_{a'}}=2\widehat{i}+3\widehat{j}+6\widehat{k}[/tex]
Similarly vector B' is written as
[tex]\overrightarrow{v_{b'}}=\widehat{i}+5\widehat{j}+3\widehat{k}[/tex]
Thus the vector dot product of the 2 vectors is obtained as
[tex]\overrightarrow{v_{b'}}\cdot \overrightarrow{v_{a'}}=(\widehat{i}+5\widehat{j}+3\widehat{k})\cdot (2\widehat{i}+3\widehat{j}+6\widehat{k})\\\\\overrightarrow{v_{a'}}\cdot \overrightarrow{v_{b'}}=1\cdot 2+5\cdot 3+3\cdot 6=35[/tex]
Explain the difference in meaning between |-3| and-3. In your explanation, be sure to include various key words.
Answer:
|-3| = 3. It indicates the distance the number -3 is from zero
Step-by-step explanation:
We have been asked that:
Explain the difference in meaning between |-3| and-3
Actually |-3| = 3. It indicates the distance the number -3 is from zero, which is 3 units in this case. Look at the attached picture. If you start from -3. Then you have to walk 3 spaces to get to the number zero. The distance in the attached picture is along a number line....
The brain volumes (cm cubed) of 50 brains vary from a low of 904 cm cubed to a high of 1488 cm cubed. Use the range rule of thumb to estimate the standard deviation s and compare the result to the exact standard deviation of 175.5 cm cubed, assuming the estimate is accurate if it is within 15 cm cubed
Answer:
standard deviation is 146 cm
Computed value of standard deviation is not near to original value.
Step-by-step explanation:
Given data:
n is number of brains = 50
low volume = 904 cm
high volume of brain = 1488 cm
As we know that range is 4 times the standard deviation so we have[tex]Range = 4\times standard\ deviation[/tex]
R = HIGH - LOW
= 1488 - 904
= 584
Therefore we have
standard deviation[tex] = \frac{R}{4}[/tex]
[tex]= \frac{584}{4}[/tex]
standard deviation is 146 cm
Original deviation is given as 175.5 cm
Computed value of standard deviation is not near to original value.
Given f(x)=2x-1
find the domain value if the range value is 4.
Answer:
5/2
Step-by-step explanation:
You want to find x when f(x) = 4
4 = 2x -1
5 = 2x . . . . . add 1
5/2 = x . . . . . divide by 2
The input corresponding to an output of 4 is 5/2.
Select the best definitions of population and sample. A sample is the group from whom information is being collected. A population is the larger group the sample represents. A population is a number that describes an entire group. A sample is a number that describes a sub-collection of that group. A sample is the complete group under study. A population is a sub-collection of members selected from the sample A sample is a group selected in such a way that each member has a known chance of being in the group. A population is the larger group the sample represents. A population is the complete group under study. A sample is the sub-collection of members of the population from which data are actually collected.
A population is the entire group of interest in a researcher's study, while a sample is a subset of this group from which data is collected. The aim is for the sample to be representative of the population to accurately draw generalizations. Effective sampling strategies and recruitment are vital for this representation.
To understand the concepts of population and sample in the context of statistics, we can differentiate between the two. Population refers to the entire group that is the focus of a researcher's study, which can be a broad group, like all adults over the age of 18 in the United States, or more specific, such as 'mid-season maturity corn plants on irrigated farms near Grand Island, Nebraska.'
A sample, on the other hand, is a subset of the population from which researchers actually collect data. It represents a smaller group selected to draw conclusions about the population. The validity of these conclusions often depends on how well the sample represents the population, aiming for the sample characteristics to match those of the population. For instance, if surveying the relative proportion of cars to trucks driving down a street, a sample observed during an uncharacteristic time of day may not provide a representative view of the overall traffic pattern.
In research, sampling strategies and recruitment techniques are important to ensure that the sample accurately reflects the population. For example, choosing individuals to participate in a study so that each has a known chance of being included makes for a better representation of the population. Researchers then analyze the sample data and attempt to generalize their findings to the entire population.
Construct a 3 x 3 matrix A, with nonzero entries, and a vector b in R3 such that b is not in the set spanned by the columns of A.
To construct a matrix A in which a vector b is not spanned by A's columns, choose b to be a vector not obtainable by linear combinations of A's columns. For example, if A is a 3 x 3 matrix with consecutive integers, then b with a sufficiently different third entry would not be in the span of A's columns.
Explanation:To construct a 3 x 3 matrix A with non-zero entries where vector b in R3 is not in the set spanned by the columns of A, it is essential to ensure that b is not a linear combination of the columns of A. First, let's define matrix A with arbitrary non-zero entries:
A = 'p'\n'\n'[1 2 3]\n'\n'[4 5 6]\n'\n'[7 8 9]\n
To find a vector b that is not spanned by the columns of A, we should choose a vector that is not a linear combination of these columns. For instance:
b = 'p'\n'\n'[1]\n'\n'[1]\n'\n'[10]\n
This vector b cannot be formed by any combination of the columns of matrix A, because there's no scalar multiples that we can multiply the columns of A by to get a z-component of 10 while also having the x and y components equal to 1. In other words, b is outside the column space of A, showing that linear independence is not achieved.
The vector [tex]\( \mathbf{b} \)[/tex] is clearly not a multiple of [tex]\( \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \)[/tex]. he matrix [tex]\[ A = \begin{bmatrix} 1 2 3 \\ 2 4 6 \\ 3 6 9 \\ \end{bmatrix} \][/tex] and vector [tex]\( \mathbf{b} \)[/tex] provided above satisfy the given conditions.
To construct a 3 x 3 matrix [tex]\( A \)[/tex] with nonzero entries and a vector [tex]\( \mathbf{b} \)[/tex] in [tex]\( \mathbb{R}^3 \)[/tex] such that is not in the [tex]\( \mathbf{b} \)[/tex] span of the columns of [tex]\( A \)[/tex], we need to ensure that [tex]\( A \)[/tex] is not full rank. A matrix [tex]\( A \)[/tex] is full rank if its columns are linearly independent and span[tex]\( \mathbb{R}^3 \)[/tex]. Since we want [tex]\( \mathbf{b} \)[/tex] not to be in the span of [tex]\( A \)[/tex] , [tex]\( A \)[/tex] must have a rank less than 3.
Let's construct [tex]\( A \)[/tex] such that two of its columns are multiples of each other, which will ensure that the matrix is rank-deficient (rank less than 3). For example:
[tex]\[ A = \begin{bmatrix} 1 2 3 \\ 2 4 6 \\ 3 6 9 \\ \end{bmatrix} \][/tex]
Here, the second column is twice the first column, and the third column is three times the first column. This implies that the columns of [tex]\( A \)[/tex] are linearly dependent, and the rank of [tex]\( A \)[/tex] is 1.
Now, let's choose a vector [tex]\( \mathbf{b} \)[/tex] that is not a multiple of the first column of [tex]\( A \)[/tex]. For instance: [tex]\[ \mathbf{b} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} \][/tex]
The vector [tex]\( \mathbf{b} \)[/tex] is clearly not a multiple of [tex]\( \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \)[/tex], which is the first column of [tex]\( A \)[/tex]. Therefore, [tex]\( \mathbf{b} \)[/tex] cannot be written as a linear combination of the columns of [tex]\( A \)[/tex] , and thus it is not in the span of [tex]\( A \)[/tex].
To verify this, we would typically attempt to solve the system [tex]\( A\mathbf{x} = \mathbf{b} \)[/tex] for [tex]\( \mathbf{x} \)[/tex]. If there is no solution, then [tex]\( \mathbf{b} \)[/tex] is not in the span of the columns of [tex]\( A \)[/tex]. In this case, since [tex]\( A \)[/tex] is rank-deficient and [tex]\( \mathbf{b} \)[/tex] is not a multiple of any column of [tex]\( A \)[/tex] , the system has no solution.
In conclusion, the matrix [tex]\( A \)[/tex] and vector [tex]\( \mathbf{b} \)[/tex] provided above satisfy the given conditions.
Travis,Jessica ,and Robin are collecting donations for the school band.Travis wants to collect 20% more than Jessica,and Robin wants to collect 35% more than Travis.If the students meet their goals and Travis collects $4,how much money did they collect in all?
Answer:
They collected $12.73 in all.
Step-by-step explanation:
This problem can be solved by a simple system of equations.
I am going to say that x is the quantity that Travis collects, y the quantity that Jessica collects and z the quantity that Robin collects.
The problems asks how much money did they collect in all?
So [tex]T = x + y + z[/tex]
Solution
The problem states that Travis wants to collect 20% more than Jessica, so:
[tex]100%x = (100%+ 20%)y[/tex]
[tex]100%x = 120%y[/tex]
[tex]x = 1.2y[/tex]
Robin wants to collect 35% more than Travis, so:
[tex]100%z = (100%+35%)x[/tex]
[tex]100%z = 135%x[/tex]
[tex]z = 1.35x[/tex]
Travis collects $4, so [tex]x = 4[/tex]. So:
[tex]x = 1.2y[/tex]
[tex]1.2y = 4[/tex]
[tex]y = \frac{4}{1.2}[/tex]
[tex]y = 3.33[/tex]
------------
[tex]z = 1.35x = 1.35(4) = 5.40[/tex]
The total is:
[tex]T = x + y + z = 4 + 3.33 + 5.40 = $12.73[/tex]
They collected $12.73 in all.
7:( 4 11 m)=56:3.2
I'm too lazy to do this....Please can someone do this for meh.
Who ever does it with steps gets a virtual hug. -w-
Answer:
0.000973236=m
Step-by-step explanation:
Given the question as;
7:(411m) = 56 : 3.2
7/411m =56/3.2
7×3.2=56×411m
(7×32)/(560×411)=m
224/230160=m
0.000973236=m
checking the answer
7: (411 *0.000973236) = 56: 3.2
7: (0.4)= 56 : 3.2
7/0.4 =56/3.2
70/4=560/32
17.5 =17.5
3p - 5 = 19
Answer when solved??
Answer: P=8
Step-by-step explanation:
3p-5=19
U turn -5 to +5
Then u add 5 to both sides, -5 and 19
3p-5=19
+5=+5
-5 and +5 cancel each other out so know it’s
3p=24
Because 19 plus 5 is 24
Now u have to get the variable by itself by dividing 3 on both sides of the equal sign 3p and 24
3 and 3 cancel each other out so now you only have p=24 but then 24 divided 3 is 8
eight increased by the product of a number and 4 is at most 20
Answer:
I am not entirely sure what you are asking, but I believe the answer is 4*2=8
Step-by-step explanation:
This is because:
8 increased by the product of a number and 4 is at most 20.
8 + p + 4 = 20
We are trying to find p.
8 +4= 12
20 - 12= 8
So I believe the answer is 8.
(If the answer is wrong, plz tell me in the comments)
Answer:
3
Step-by-step explanation:
eight increased by the product of a number and 4 is at most 20
these implies
8 + a x4 = 20
a is the number whose product with 4 is increased by eight to give at most 20
8 + a x4 = 20
8 + 4a = 20
subtract 8 from both sides
4a = 20 -8
4a= 12
divide both sides by 4
a = 12/4 = 3
the number is 3
translate into algebraic expression 10 more than a number
Answer:
x+10
Step-by-step explanation:
An algebraic expression is an expression which consist of variables(whose values are not fixed like in the form of x,y,a,...), the constants and operators (like +,×,±,-,≤,≥,=,...).
Now for this question we have to give a an algebraic expression for 10 more than a number.
Let the number be x.
We have to show a relation of 10 more than the number. Thus are algebraic expression is of the form x+10.
Algebraic expression: x+10
where x is our variable
+ is our operator and
10 is a constant.
If we have a 90% confidence interval for a population parameter, then A. 10% of the time, the population parameter will lie outside of the interval. B. 10% of the time, the population parameter will be lower than the LCL. C. 10% of the time, the population parameter will be higher than the UCL. D. All of the above will hold.
Answer: A. 10% of the time, the population parameter will lie outside of the interval.
Step-by-step explanation:
If we have [tex]b\%[/tex] confidence interval is that we are [tex]b\%[/tex] certain that it contains the true population parameter in it.
Similarly , if we have a 90% confidence interval for a population parameter, then we are 90% certain that it contains the true population parameter in it.
i.e. 10% not certain that it contains the true population parameter in it.
i.e. 10% of the time, the population parameter will lie outside of the interval.
Negate: "In every good book there is a plot twist or surprise ending."
Answer:
"There exists a good book that does not have a plot twist and does not have a surprise ending".
Step-by-step explanation:
We negate the universal quantifier "for all" or equivalently "In every" using the existential quantifier "There exists". So, we negate "In every good book" as "There exists a good book". In the other hand, we have the propositions
P: there is a plot twist
Q: there is a surprise ending,
and the conjunction
P ∨ Q. We negate this conjunction using the De Morgan's Laws as
¬(P∨Q) = ¬P∧¬Q
i.e., does not have a plot twist and does not have a surprise ending. Therefore, we negate "In every good book there is a plot twist or surprise ending" as "There exists a good book that does not have a plot twist and does not have a surprise ending".
A company asks an investor for an investment of $950,000 in exchange for 25% equity in the business. What is the implied valuation of the business?
Answer:
$2,850,000
Step-by-step explanation:
Data provided in the question:
Investment amount asked for by the company = $950,000
Exchange of equity = 25%
Now,
Equity exchanged = [tex]\frac{\textup{Amount invested}}{\textup{Post money evaluation}}\times100[/tex]
or
Post money evaluation = [tex]\frac{\textup{950,000}}{\textup{25}}\times100[/tex]
or
Post money evaluation = $3,800,000
Therefore,
Implied valuation = Post money evaluation - Amount invested
or
Implied valuation = $3,800,000 - $950,000 = $2,850,000
translate phrases into algebraic expressions, the difference between the cube of a number and the square of the number
Answer:
[tex]x^3-x^2.[/tex]
Step-by-step explanation:
The word "difference" represents a subtraction. Then the algebraic expression will be of the form a-b.
Now, the difference is between the cube of a number and the square of the number, then let's call the number x. The square of the number is raise to two. Then the square of the number is [tex]x^2[/tex].
The cube of the number is raise to three. Then the cube of the number is [tex]x^3[/tex].
So, the difference between the cube of a number and the square of the number (we are talking about the same number in the square and the cube) is [tex]x^3-x^2[/tex].
Let x, y be integers. What possible values can x^2 + y^2 take in Z4
Answer:
0, 1 or 2.
Step-by-step explanation:
An integer x in Z4 is either equal to [0], [1], [2] or [3] (as Z4 is made only of the remainders we can get when dividing an integer by 4).
If x was equal to [0] in Z4, then x^2 = [0]*[0]=[0] in Z4.
If x was equal to [1] in Z4, then x^2 = [1]*[1]=[1] in Z4.
If x was equal to [2] in Z4, then x^2 = [2]*[2]=[4]=[0] in Z4 (as 4 and 0 are the same in Z4, given that both numbers leave a remainder of 0 when divided by 4).
If x was equal to [3] in Z4, then x^2 = [3]*[3]=[9]=[1] in Z4 (as 9 and 1 are the same in Z4, given that both numbers leave a remainder of 1 when divided by 4).
Therefore, in Z4 x^2+y^2 is either a sum of the form [0]+[0], or [0]+[1], or [1]+[0], or [1]+[1], which means we can only get either [0], [1] or [2].
Given the proposition,
P(n): 1 + 2 + 2^2 + 2^3 + . . . + 2n = 2n+1 - 1, n = 0, 1, 2, . . .
Find the values of:
P(0)
P(1)
P(2)
P(n+1)
Answer and Explanation:
Given : [tex]P(n): 1 + 2 + 2^2 + 2^3 + . . . + 2^n = 2^{n+1} - 1[/tex], n=0,1,2,..
To find : The values of following expression ?
Solution :
The function is [tex]P(n)=2^{n+1} - 1[/tex]
1) Value of P(0),
[tex]P(0)=2^{0+1} - 1[/tex]
[tex]P(0)=2^{1} - 1[/tex]
[tex]P(0)=2 - 1[/tex]
[tex]P(0)=1[/tex]
2) Value of P(1),
[tex]P(1)=2^{1+1} - 1[/tex]
[tex]P(1)=2^{2} - 1[/tex]
[tex]P(1)=4- 1[/tex]
[tex]P(1)=3[/tex]
3) Value of P(2),
[tex]P(2)=2^{2+1} - 1[/tex]
[tex]P(2)=2^{3} - 1[/tex]
[tex]P(2)=8- 1[/tex]
[tex]P(2)=7[/tex]
4) Value of P(n+1),
[tex]P(n+1)=2^{n+1+1} - 1[/tex]
[tex]P(n+1)=2^{n+2} - 1[/tex]