Quiz 6

Consider the 2 x 2 matrix [ 6 4 ] A = [ 2 4 ] The eigenspace of λ = 2 is spanned by what vector?

(1,-1)

Consider the 2 x 2 matrix [ 5 - 2 ] A = [ 1 2 ] The eigenspace of λ = 3 is spanned by what vector? (1, 5) (1, 1) (1, 2) (- 2, 1) (2, - 1) (5, 2)

(1,1)

A 2 x 2 matrix A has trace 6 and determinant 8. Its eigenvalues are 4 and 8 6 and 8 2 and 6 4 and 6 2 and 8 2 and 4

2 and 4

The eigenvalues of the 2 x 2 matrix [ 3 9 ] A = [ 0 4 ] are 3 and 9 4 and 9 3 and 4 3, 4, and 9

3 and 4

Let A be a 5 x 7 matrix with rank 3. Then dim Nul A is

4

Let A be a matrix with echelon form U. Which of the following statements is not correct? A and U have the same row space A and U have the same column rank A and U have the same column space A and U have the same row rank

A and U have the same column space

Elementary row operations cannot change Dependence relations between the columns of a matrix The column space of a matrix Dependence relations between the rows of a matrix

Dependence relations between the columns of a matrix

If a square matrix A is diagonalizable then P A P = D where P is an invertible matrix and D is a diagonal matrix

F

Le A be a square matrix. If x is in Nul (A - λ I) then (λ, x) is an eigenpair for A.

F

Let A be a square matrix with eigenvalue λ. The eigenspace of λ is the nullspace of A.

F

Let A be a square matrix with eigenvalue λ.Then A - λ I is invertible.

F

Let B and C be ordered bases of Rn. Let PB be the change-of-coordinates matrix from basis B to the standard basis E, and let PC be the change-of-coordinates matrix from basis C to the standard basis E. Applying GJ to [ PB | PC ] produces [ I | PB → C ] where I is the identity matrix and PB → C is the change-of-coordinates matrix from basis B to basis C.

F

Matrices that have the same eigenvalues are always similar

F

The characteristic polynomial of a square matrix is obtained by subtracting λ from each of its entries and then taking the determinant

F

The eigenspace of an eigenvalue always has dimension one

F

Let A be a square n x n matrix. Which of the following statements is not correct? If A has eigenvalue λ then Nul (A - λ I) is a non-trivial subspace of Rn The eigenvalues of A are the zeros of det (A - λ I) det (A - λ I) is a polynomial in λ of degree n If A has eigenvalue λ then A - λ I is singular If A has eigenpair (λ, x) then A x = λ x with λ a non-zero scalar and x a vector in Rn If x belongs to Nul (A - λ I) with x a non-zero vector then A has eigenpair (λ, x) If A has eigenvalue λ then det (A - λ I) = 0 If A has eigenpair (λ, x) then x belongs to Nul (A - λ I) If A has eigenpair (λ, x) then (λ, a x) is also an eigenpair for any non-zero scalar a

If A has eigenpair (λ, x) then A x = λ x with λ a non-zero scalar and x a vector in Rn

Let A be a matrix with echelon form U​. Then dim Nul A is equal to the number of Non-zero rows in U Pivot columns in U Non-pivot columns in U Zero rows in U

Non-pivot columns in U

A square n x n matrix A is invertible if and only if Col A is all of Rn

T

A square n x n matrix A is invertible if and only if Nul A is the trivial subspace of Rn

T

A square n x n matrix A is invertible if and only if its columns form a basis for Rn

T

An invertible n x n matrix is always a change-of-coordinates matrix for some pair of ordered bases B and C of Rn

T

Let A be a square matrix with eigenpair (λ, x). Then x is in Nul (A - λ I).

T

Let A be a square matrix with eigenpairs (3, u) and (5, v). Then u and v are linearly independent.

T

Row equivalent matrices have the same row space

T

Similar matrices always have the same eigenvalues

T

Let V be a vector space with dim V = 2. Let B = {b1, b2} and C = {c1, c2} be two ordered bases for V with b1 = 3 c1 + 2 c2 and b2 = 5 c1 - 2 c2. Let PB → C be the change-of-coordinates matrix from basis B to basis C and let PC → B be the change-of-coordinates matrix from basis C to basis B. Then The columns of PC → B are (3, 2) and (5, - 2) The rows of PB → C are [3, 2] and [5, - 2] The rows of PC → B are [3, 2] and [5, - 2] The columns of PB → C are (3, 2) and (5, - 2)

The columns of PB → C are (3, 2) and (5, - 2)

Let V be a vector space with dim V = n. Let B and C be ordered bases for V. Let x be a vector in V. Let [x]B and [x]C be the coordinate vectors of x relative to the bases B and C, respectively. Let PB → C be the change-of-coordinates matrix from basis B to basis C. Which of the following statements is not correct? You Answered The inverse of PB → C is the change-of-coordinates matrix from basis C to basis B PB → C is an n x n matrix The columns of PB → C are the coordinate vectors for the elements of C relative to B [x]C = PB → C [x]B

The columns of PB → C are the coordinate vectors for the elements of C relative to B

Let A be a matrix with echelon form U. Which of the following is not equal to the others? The row rank of A The dimension of the null space of A The number of non-zero rows in U The column rank of A

The dimension of the null space of A

Let A be a matrix with echelon form U. Which of the following statements is correct? The non-zero columns of U form a basis for the null space of A The non-zero columns of U form a basis for the column space of A The non-zero rows of U form a basis for the row space of A

The non-zero rows of U form a basis for the row space of A

Let A be a matrix with echelon form U​. Which of the following is a basis for the column space of A? The pivot columns of A The pivot columns of U The non-pivot columns of A The non-pivot columns of U

The pivot columns of A

Elementary row operations cannot change Dependence relations between the rows of a matrix The row space of a matrix The column space of a matrix

The row space of a matrix

Let x be a vector in R2. Let [x]B be the coordinate vector of x relative to the ordered basis B = {b1, b2} for R2. Let [x]C be the coordinate vector of x relative to the ordered basis C = {c1, c2} for R2. Let PB = [b1, b2] (the matrix whose columns are b1 and b2) and PC = [c1, c2] (the matrix whose columns are c1 and c2). Let PB → C be the change-of-coordinates matrix from basis B to basic C. Which of the following statements is not correct? [x]C = PB → C [x]B x = PC [x]C [x]C = PC PB [x]B PB [x]B = PC [x]C

[x]C = PC PB [x]B

Let A be a matrix with two columns, a and b, and let S be its column space. Which of the following statements is not correct? S = {r a + s b : r and s are real} dim S = 2 S is a vector space S = span(a, b)

dim S = 2