Linear Algebra Answers

Find the Square matrix M order of two such that M square of 2 - 2M= (-1 0) 2x2
6 3

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 0 1
A = 2 -1 4
3 -1 5

rank (A) =

2.
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.

A = 1 -1 6 0
8 1 0 0
-1 6 12 0

rank (A) =

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) =