Linear Algebra Answers

Suppose A is a square matrix such that det(A) = 2 and det(3A the power of t) = 18 then find the order of matrix A

1. Consider the basis S = {v1, v2} for R2, where v1 = (− 2, 1) and v2 = (1, 3), and let T:R2 → R3 be the linear transformation such that


T(v1) = (− 1, 2, 0) and T(v2) = (0, − 3, 5)


Find a formula for T(x1, x2), and use that formula to find T(2, − 3).


Give exact answers in the form of a fraction.


2. Consider the basis S = {v1, v2, v3} for R3, where v1 = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 0), and let T:R3 → R3 be the linear operator for which


T(v1) = (3, − 1, 6), T(v2) = (4, 0, 1), T(v3) = (− 1, 7, 1)


Find a formula for T(x1, x2, x3), and use that formula to find T(3, 6, − 1).


3. Let v1, v2, and v3 be vectors in a vector space V, and let T:V → R3 be a linear transformation for which


T(v1) = (1, − 1, 2), T(v2) = (0, 3, 2), T(v3) = (− 3, 1, 2)


Find T(4v1 − 5v2 + 6v3).

1. Find a 3 × 3 matrix A that has eigenvalues λ = 0, 6, − 6 with corresponding eigenvectors


0 0 0

1 , -1 , 1

-1 1 1


2. Let Upper T ⁢ colon Upper M Subscript 22 Baseline ⁢ right-arrow Upper M Subscript 22 be the dilation operator with factor k = 3.


(a) Find Upper T left-parenthesis Start 2 By 2 Matrix 1st Row 1st Column 1, 2nd Column 4 2nd Row 1st Column -7, 2nd Column 4 End Matrix right-parenthesis = ????


(b) Find the rank and nullity of T .


3. Let Upper T ⁢ colon Upper P Subscript 2 Baseline ⁢ right-arrow Upper P Subscript 2 be the contraction operator with factor k ⁢ equals StartFraction 1 Over 6 EndFraction Number .


(a) Find Upper T left-parenthesis 1 ⁢ plus 6 x ⁢ plus 12 x Superscript 2 Baseline right-parenthesis Number .


(b) Find the rank and nullity of T.

1. Find the Eigen values of


0 1 0

A= 0 0 1

216k^3 -108k^2 18k


lambda 1 = ?

lambda 2= ?

lambda 3= ?


2. Using the fact that the matrix


0 0 0 ... 0 -c subscript 0

1 0 0 ... 0 -c subscript 1

0 1 0 ... 0 -c subscript 2

... ... ... ... ... ...

0 0 0 ... 1 -c subscript n-1


has the characteristic polynomial


p(λ) = c0 + c1λ + ⋯ + cn − 1λn − 1 + λn


find a matrix with the characteristic polynomial


p(λ) = 1 − 5λ + λ2 + 6λ3 + λ4

C03


1. Write the characteristic equation of A = 9 1

-2 -3


2. Use the function to find the image of v= ( 1 4 9).

T(v1, v2, v3)=(v2-v1, v2 is not equal to v2, 2v3)


3a. Determine whether or not T R^2 arrow R is a linear transformation.

T(a,b)= a^2.


3b. Determine whether or not T R^3 arrow R is a linear transformation.

T(a,b)= a^3


4. Let A = 6 16 Show that -2 is an eigenvector with the corresponding eigenvalue of -2.

-1 -4 1

C02


1. Determine whether or not the set W = {(a + 2, a, 0) : a is a real number } is a subspace of R^3 with the standars operations.


2. Let V= R^2, and define addition and scalar multiplication as follows:

u+v= (u1, u2) + (v1, v2) = (u1+v1+1, u2+v2+2)

ku= k(u1,u2)=(ku1,-ku2)

Let u=(2,1) and v=(4,-1). Compute the value of 2u+v under the given operations.

(Note that V is not a vector space.)



3. Use a determinant to prove that the vectors (6,3,2), (3, 6, 0) and (0, 0, 2) form a basis of R^3.


4. Find the basis of the row space of the matrix.


1 -2 3 9

A= 0 1 3 5

0 -1 4 9

C01


1.If Matrix A is 4 x 2, B is 3 x 4, C is 2 x 4, and D is 4 x 3, what is the size of the given expression.


A^TD + CB^T


2. Find the value of the k that makes the system 15 3 6 inconsistent.

-10 k 9


3.What is the second row of the product AB?


0 2 3 2 1 7

A = 5 4 8 B = 6 3 2

9 7 2 2 9 7


4. A is a 3 x 3 matrix and det(A)=7. what is det(2A).

1.Various advanced texts in linear algebra prove the following determinant criterion for rank:


The rank of a matrix A is r if and only if A has some r × r sub matrix with a nonzero determinant, and all square sub matrices of larger size have determinant zero.


(A sub matrix of A is any matrix obtained by deleting rows or columns of A. The matrix A itself is also considered to be a sub matrix of A.) Use this criterion to find the rank of the matrix.


1 4 0

3 12 -1

rank (A) =



rank (A) =

1. For purposes of this exercise, let us define an " X -matrix" to be a square matrix with an odd number of rows and columns that has 0's everywhere except on the two diagonals, where it has 1's. Find the rank and nullity of the following X -matrix.


close and open parenthesis 1 0 1

0 1 0

1 0 1


rank (X) =

nullity (X) =

1. For purposes of this problem, let us define a "checkerboard matrix" to be a square matrix A = [aij] such that


a Subscript i j Baseline equals Start Layout left-brace 1st Row 1st Column 1 if i+j is even 2nd Row 1st Column 0 if i+j is odd End Layout


Find the rank and nullity of the following checkerboard matrix.


The 6 × 6 checkerboard matrix.


a. Rank (A) =

b. nullity (A) =


2. For purposes of this problem, let us define a "checkerboard matrix" to be a square matrix Upper A equals left-bracket a Subscript i j Baseline right-bracket such that


a Subscript i j Baseline equals Star tLayout left-brace 1st Row 1st Column -2 if i plus j is even 2nd Row 1st Column 0 if i plus j is odd End Layout


Find the rank and nullity of the n times n checkerboard matrix for n greater-than-or-equal-to 2.


a. Rank (A) =

b. nullity (A) =