Linear Algebra Questions and Answers

Math - Others
Linear AlgebraProve that nullity (A) = nullity(A^T) if and only if A is a square matrix.
![1. Compute the following by using
2
A=
10
^- [3 ]. B-[282]. and v-[-]
4
23
a) A + 2B and (A + 2B)T
b) (A + 2B)v and ((A + 2B)v)
c) VT (A + 2B)](https://media.kunduz.com/media/sug-question/raw/80206970-1659652424.1791008.jpeg?w=256)
Math - Others
Linear Algebra1. Compute the following by using
2
A=
10
^- [3 ]. B-[282]. and v-[-]
4
23
a) A + 2B and (A + 2B)T
b) (A + 2B)v and ((A + 2B)v)
c) VT (A + 2B)

Math - Others
Linear AlgebraLena spent a total of $60 at the grocery store. Of this amount, she spent $42 on fruit. What percentage of the total did she spend on fruit?

Math - Others
Linear AlgebraFor :9-11, use the points given to find the slope between them. (Answers might be fractions!)
9. (-3,4) and (-1,-2)
10. (7,9) and (2, -1)
11. (-21,9) and (19,-4)

Math - Others
Linear AlgebraFind the product of the complex numbers. Leave your answer in polar form.
Z₁ = 5( cos 65° + i sin 65°)
Z₂ = 9( cos 50° + i sin 50°)
Choose the correct answer below.
Z₁ Z2 = 14(cos 3250° + i sin 3250°)
Z₁22=45(cos 115° sin 115°)
Z₁ Z2 = 14(cos 15° + i sin 15°)
Z₁ Z2 =45(cos 115° + i sin 115°)

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Linear Algebra18. Let A be a 3x3 real nonsingular matrix such that A^-1 = A. Characterize its eigenvalues.
1. -2, 1, or -1
2. 1 or 2
3. -1 or -2
4. 1 or -1
5. -1, 1, or 2
6. None of the above.

Math - Others
Linear Algebraã=< -1, -3> and b= <5, 4>
Represent a + b using the parallelogram method.
Use the Vector tool to draw the vectors, complete the parallelogram method, and draw a + b.

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Linear AlgebraThe high temperature in Jackson, WY, on July 13 was 80°F. Use the formula, C = 5/9 (F – 32), where C is Celsius degrees and F is Fahrenheit degrees, to convert 80°F to Celsius degrees. Round to the nearest tenth of a degree.

Math - Others
Linear AlgebraFind the 11th term of the geometric sequence shown below.
7x⁷, 28x¹¹, 112x¹⁵, ...
Answer: _____________

Math - Others
Linear AlgebraThe first three terms of a geometric series are sinθ cosθ and 0.5 where θ is a constant
(a) Show that 2sin^θ + sinθ -2 = 0
Given that lies in the interval 90° < θ < 180°
(b) find the value of θ giving your answer to one decimal place.
(c) Hence prove that this series is convergent.
(d) Find, two significant figures, S∞

Math - Others
Linear AlgebraSolve the following system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent
{-x +y +z = -2, -x +3y -3z = -16, 7x -5y -11z = 0
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The solution is (___+____+____). (Simplify your answers.)
B. There are infinitely many solutions. The solution can be written as {(y, z)|x = ____ .y = ____ .z is any real number}. (Simplify your answers. Type expressions using z as the variable.)
C. There are infinitely many solutions. The solution can be written as {(x, y, z)l x = ____ .y is any real number, z is any real number} (Simplify your answer. Type an expression using y and z as the variables.)

Math - Others
Linear AlgebraA circular field in a park will be planted with sod. Sod cost $3 per square yard
• If the diameter of the circle is 22 yards, what is the radius_________.
• Use the formula A =πr^2
• A = ( _______)(____________)^2
• A = (________)(____________)
• A =__________ square yards (round to nearest whole number)
• Find the cost of the sod needed
• cost equals ($___________)(_________)
• The sod will cost $ _____________.
Blank 1:
Blank 2:
Blank 3:
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

Math - Others
Linear Algebra1 X
X 1
Calculate a) Spectral decomposition of the matrix
b) Rank-1 approximation of the matrix

Math - Others
Linear AlgebraThe diameter of the earth is 7,917.5 miles. Calculate the volume of the earth.

Math - Others
Linear AlgebraExamine the function f(x)= x/6 -2
Answer the following questions.
What is the slope of the function? (**HINT** What number times x will equal x/6 ?)
Provide five input values for x that will generate integer outputs for the function.
Make sure two of your input values are negative. Since m=1/6, any multiple of 6 will generate an integer as the output.
![Consider the matrix
A= 1 2 4 -3
2 4 3 1
1 0 2 -6
0 1 1 2
Form the augmented matrix [ A| I ] , where I is the 4 X 4 identity matrix . Use Matlab to reduce [ A | I ] to reduced row echlon form . Write A-1](https://media.kunduz.com/media/sug-question/raw/79137689-1659644527.9364839.jpeg?w=256)
Math - Others
Linear AlgebraConsider the matrix
A= 1 2 4 -3
2 4 3 1
1 0 2 -6
0 1 1 2
Form the augmented matrix [ A| I ] , where I is the 4 X 4 identity matrix . Use Matlab to reduce [ A | I ] to reduced row echlon form . Write A-1

Math - Others
Linear AlgebraIn a recent year a country had a GDP of $575,000,000 and a per capita GDP of $2,500. Based on this information, what was the population of that country in that year?

Math - Others
Linear AlgebraIn a shop window, a jogging outfit is marked “$99.45 —was $127.50." What is the rate of discount?

Math - Others
Linear Algebra"A polynomial f(r) over an field is irreducible if and only f(x + 1) is irreducible." Prove or disprove.

Math - Others
Linear AlgebraYou want to invest $1,000. Which growth curve would yield the largest principal 5 years from now? 10 years? 20 years?
(a) yt = 1,000e^0.040t
(b) yt = 1,000 + 48t
(c) yt = 1,000+ 9t+5t²
Year curve a curve b curve c Largest principal
5
10
15

Math - Others
Linear AlgebraA mapping 7: R² → R² is expressed by the transformation matrix below,
A = 2 5
-3 7
(a) State the condition for an orthogonal transformation. Is T an orthogonal transformation ? Give your reason.
(b) Find the image of the given vector below using the transformation matrix A.
U = 5
-1

Math - Others
Linear AlgebraCompute Aut(Q(√2, √3)) with justification. (Give both its size and group structure.)

Math - Others
Linear AlgebraFind an equation of a generating curve of the surface of revolution
y^2 + z² - 4x = 0.

Math - Others
Linear AlgebraGiven v= -7i+ 5j and w=-i-j.
a. Find projwv.
b. Decompose v into two vectors v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w.
![Compute the following by using
A =[2 -1 5] b=[1 0 -2 ] and v=[1]
[3 4 1] [2 3 4] [-2]
[5]
a) A + 2B and (A + 2B)^T
b) (A + 2B)v and ((A + 2B)v)^T
c) v^t (A + 2B)^T](https://media.kunduz.com/media/sug-question/raw/80207464-1659638365.122351.jpeg?w=256)
Math - Others
Linear AlgebraCompute the following by using
A =[2 -1 5] b=[1 0 -2 ] and v=[1]
[3 4 1] [2 3 4] [-2]
[5]
a) A + 2B and (A + 2B)^T
b) (A + 2B)v and ((A + 2B)v)^T
c) v^t (A + 2B)^T

Math - Others
Linear AlgebraLet u = 3i+ 3j, v=3i-j, and w= -5j.
Find the specified scalar or vector.
5u (3v-4w)

Math - Others
Linear AlgebraFind the first six terms of the sequence.
a₁ = 4, a₂ = 5; for n ≥3, aₙ = aₙ - 1 + aₙ - 2

Math - Others
Linear AlgebraAn urban community would like to show that the incidence of breast cancer is higher in their area than in a nearby rural area. (PCB levels were found to be higher in the soil of the urban community.) If it is found that 20 of 200 adult women in the urban community have breast cancer and 10 of 150 adult women in the rural community have breast cancer, can we conclude at the 0.05 level of significance that breast cancer is more prevalent in the urban community?
What is the type of test to be used?
(A) z-test, left tailed
(B)t-test, two tailed
(C) t-test, left tailed
(D)z-test, two tailed
(E)t-test, right tailed
(F)z-test, right tailed

Math - Others
Linear AlgebraLet be the subspace
W={(x +y, y, 2x−y)x, y ∈ R³}
with usual inner product, determine an orthogonal basis of W, the orthogonal complement of W and the projection of the vector u=(1,-3,-2) on W

Math - Others
Linear AlgebraConstruct a consistent, unstable multistep method of order 2, other than
Yn = -4yn-1 + 5yn-2 +4hfn-1+2hfn-2.

Math - Others
Linear AlgebraA study concluded that following 7 simple health rules can extend a man's life by 11 years on the average and a woman's life by 7 years. These 7 rules are as follows: no smoking, get regular exercise, use alcohol only in moderation, get 7 to 8 hours of sleep, maintain proper weight, eat breakfast, and do not eat between meals. In how many ways can a person adopt 5 of these rules to follow
(a) if the person presently violates 4 of the 7 rules?
(b) if the person never smokes and always gets between 7 and 8 hours of sleep?
(a) There are___ways.

Math - Others
Linear AlgebraGiven the vector u=(1, -3), find the magnitude and angle in which the vector points (measured
counterclockwise from the positive x-axis, 0≤ θ< 2π)
counterclockwise
||ū|| =
θ=

Math - Others
Linear Algebra1. Given an equation of the second degree
3x² + 12xy + 8y² - 30x - 52y + 23 = 0
a. Use translation and rotation to transform the equations in the
simplest standard form
b. Draw the equation curve
c. Determine the focal point of the equation

Math - Others
Linear AlgebraSuppose V is finite-dimensional with dim V>0, and suppose W is infinite-dimensional. Prove that L (V, W) is infinite-dimensional.

Math - Others
Linear AlgebraFind the rank and nullity of the given matrix:
1 0 1 -1 6
2 -1 5 -1 7
-1 1 -4 1 -3
0 1 -3 1 1

Math - Others
Linear AlgebraConsider three vectors u, v and w has the following relations:
(m² - 4) u + (m² - 3m + 2) v + (m² - 3m – 4) w = 0
(A) (l : 1) if m= 3, what can you conclude about u, v and w?
(B) (K : 1, l : 1) Determine the value(-s) of m, if u, v and w spans R³.
(C) (K : 1, l : 1) Determine the value(-s) of m, if u, v and w spans R².

Math - Others
Linear AlgebraDetermine the values of r for which v = 1 is in the span of S = 1 , 1
r 2 -2
-4 -1 2

Math - Others
Linear AlgebraAssume the vector space is defined by the following: {(x,y,z) |y=x², z=2x|}
Does the definition satisfy the vector space axioms for addition and scalar multiplication?

Math - Others
Linear AlgebraLet A be a 4 x 4 diagonalizable matrix. Which of these are always true about A?
(I) A is similar to a diagonalizable matrix.
(II) All the eigenvalues of A are not equal to 0.
(III) The set of all eigenvectors of A spans R⁴.
A. Only (I) and (III).
B. Only (I) and (II).
C. Only (I).
D. All of (I), (II) and (III).
E. None of these.

Math - Others
Linear AlgebraLet f be: R²→R given by f(x,y) = (y -x²) (y – 2x²)
Show that f has no local minimum at (0,0), but that every restriction flg for every line G⊂R² through the origin, has a local minimum at (0,0).

Math - Others
Linear Algebra(10%) Let T be the function that maps R² into R² such that T(υ) = projᵥυ, where ν= (1, 1). (a) Find T(x, y). (b) Find T(5, 0). (c) Prove that T is a linear transformation form R² into R².

Math - Others
Linear AlgebraPlease fill in the following table with yes/no to indicate which of these decompositions are applicable for A and B: LU without permutation, QR, SAS-¹, QAQᵗ, and UΣVᵗ?
Decomposition Matrix A Matrix B
LU without permutation
QR
SAS-¹
QAQᵗ
UΣVᵗ
0 11 1 1 1
A= 0 1 0 B=1/3 1 1 1
1 0 0 1 1 1

Math - Others
Linear AlgebraA man pushes a lawn mower with a force of 30 i exerted at an angle of 30° to the ground. Find the horizontal and vertical components of the force, (Assume that the horizontal direction is positive and the vertical direction is negative)

Math - Others
Linear AlgebraSally and Sammy are on a merry-go-round. Sally sits twice as far from the center as Sammy. Sammy's tangential velocity would be...
A. the same as Sally's
B. 1/2 of the tangential velocity of Sally
C. twice the tangential velocity of Sally

Math - Others
Linear AlgebraA comet has an orbital radius twice that of the earth. What is the period of the comet in years?

Math - Others
Linear AlgebraFind the x- and y-intercepts.
y = |x + 1|-5
Select one:
a. x-intercepts: (4, 0) and (-6, 0); y-intercept: (0, -5)
b. x-intercept: (4, 0); y-intercept: (0, -4)
c. x-intercepts: (4, 0) and (-6, 0); y-intercept: (0, -4)
d. x-intercept: (-6, 0); y-intercept: (0, -5)

Math - Others
Linear AlgebraConvert the rectangular equation to a polar equation.
x² + y² = 16
The polar equation of x² + y² = 16 is


Math - Others
Linear AlgebraEvaluate the expression (-4-4i)(1 - 4i) and write the result in the form a + bi.