Linear Algebra Questions and Answers

Prove 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)

Lena 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?

For :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)

Find 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°)

18. 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.

ã=< -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.

The 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.

Find the 11th term of the geometric sequence shown below. 7x⁷, 28x¹¹, 112x¹⁵, ... Answer: _____________

The 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∞

Solve 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.)

A 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: Blank 4: Blank 5: Blank 6: Blank 7: Blank 8: Blank 9:

How much larger is a 5/8 inch socket than a 17/ 32 inch socket?

1 X X 1 Calculate a) Spectral decomposition of the matrix b) Rank-1 approximation of the matrix

The diameter of the earth is 7,917.5 miles. Calculate the volume of the earth.

Examine 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

In 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?

In a shop window, a jogging outfit is marked “$99.45 —was $127.50." What is the rate of discount?

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

You 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

A 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

Compute Aut(Q(√2, √3)) with justification. (Give both its size and group structure.)

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

Given 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

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

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

An 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

Let 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

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

A 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.

Given 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=

1. 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

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

Find 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

Consider 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.

Determine the values of r for which v = 1 is in the span of S = 1 , 1 r 2 -2 -4 -1 2

Assume 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?

Let 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.

Let 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).

(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.

Please 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

A 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)

Sally 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

A comet has an orbital radius twice that of the earth. What is the period of the comet in years?

Find 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)

Convert the rectangular equation to a polar equation. x² + y² = 16 The polar equation of x² + y² = 16 is

Rewrite the Cartesian equation y = 2x² as a polar equation. r(0) =

Evaluate the expression (-4-4i)(1 - 4i) and write the result in the form a + bi.