02 Matrix Algebra

Question 1

If A is an m x n matrix, and if B is an n x p matrix with columns b1, …, bp, then the product AB is the m x p matrix whose columns are ______. That is,

AB = ______

Answer 1

If A is an m x n matrix, and if B is an n x p matrix with columns b1, …, bp, then the product AB is the m x p matrix whose columns are Ab1, …, Abp. That is,

AB = A[b1 b2 … bp] = [Ab1 Ab2 … Abp]

Question 2

Each column of AB is a __ of the columns of A using ____ from the corresponding column of B

Answer 2

Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B

Question 3

row_i(AB) = __

Answer 3

row_i(AB) = row_i(A) B

Question 4

The cancellation laws do ___ for matrix multiplication. That is, if AB = AC, then it is ___ in general that B = C.

Answer 4

The cancellation laws do not hold for matrix multiplication. That is, if AB = AC, then it is not true in general that B = C.

Question 5

If a product AB is the zero matrix, you ___ conclude in general that either A = 0 or B = 0.

Answer 5

If a product AB is the zero matrix, you cannot conclude in general that either A = 0 or B = 0.

Question 6

(A + B)^T​ =

Answer 6

(A + B)^T​ = A^T + B^T

Question 7

(AB)^T =

Answer 7

(AB)^T​ = B^TA^T

Question 8

The transpose of a product of matrices equals the product of their transposes _____

Answer 8

The transpose of a product of matrices equals the product of their transposes in the reverse order.

Question 9

I (identity matrix) =

Answer 9

I = A^-1A = AA^-1

Question 10

If ____, then A is invertible and

A^-1 = ___

Answer 10

If ad - bc != 0, then A is invertible and

Question 11

If A is an invertible n x n matrix, then for each b in Rn, the equation Ax = b has the ____ solution _____.

Answer 11

If A is an invertible n x n matrix, then for each b in Rn, the equation Ax = b has the unique solution x = A^-1​b.

Question 12

If A and B are n x n invertible matrices, then ____

Answer 12

If A and B are n x n invertible matrices, then so is AB

Question 13

(AB)^-1 =

Answer 13

(AB)^-1 = B^-1A^​-1

Question 14

(A^T)^-1 =

Answer 14

(A^T)^-1 = (A^-1)^​T

Question 15

The product of n x n invertible matrices is invertible, and the inverse is the product of their inverses ___

Answer 15

The product of n x n invertible matrices is invertible, and the inverse is the product of their inverses in the reverse order

Question 16

An elementary matrix is one that is obtained by ____.

Answer 16

An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix.

Question 17

If an elementary row operation is performed on an m x n matrix A, the resulting matrix can be written as EA, where the m x m matrix E is created by ____.

Answer 17

If an elementary row operation is performed on an m x n matrix A, the resulting matrix can be written as EA, where the m x m matrix E is created by performing the same row operation on I_m​.

Question 18

____ is the elementary matrix of the same type that transforms E back into I.

Answer 18

Each elementary matrix E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I.

Question 19

An n x n matrix A is invertible if and only if A is row equivalent to __, and in this case, any sequence of elementary row operations that reduces A to __ also transforms ___.

Answer 19

An n x n matrix A is invertible if and only if A is row equivalent to I_n, and in this case, any sequence of elementary row operations that reduces A to I_n also transforms I_ninto A^-1​.

Question 20

Algorithm for finding A^-1

Answer 20

Row reduce the augmented matrix [A I]. If A is row equivalent to I, then [A I] is row equivalent to [I A^-1]. Otherwise, A does not have an inverse.

Question 21

The Invertible Matrix Theorem

Answer 21

Let A be square n x n matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false.

• A is an invertible matrix.
• A is row equivalent to the n x n identity matrix.
• A has n pivot positions.
• The equation Ax = 0 has oinly the trivial solution.
• The columns of A form a linearly independent set.
• The linear transformation x |-> Ax is one-to-one.
• The equation Ax = b has at least one solution for each b in Rn.
• The columns of A span Rn.
• The linear transformation x|-> Ax maps Rn onto Rn.
• There is an n x n matrix C such that CA = I.
• There is an n x n matrix D such that AD = I.
• A^T is an invertible matrix.

Question 22

Let A and B be square matrices. If AB = I, then A and B are ____

Answer 22

Let A and B be square matrices. If AB = I, then A and B are both invertible, with B = A^-1 and A = B^-1.

Question 23

invertible =

noninvertible =

Answer 23

invertible = nonsingular

noninvertible = singular

Question 24

If the columns of a 4 x 3 matrix are linearly independent, ____ solutions to Ax = b.

Answer 24

If the columns of a 4 x 3 matrix are linearly independent, we cannot use the IMT to conclude anything about the existence or nonexistence of solutions to equations of the form Ax = b.

Question 25

Translating an object on a screen does not correspond directly to matrix multiplication because _____. The standard way to avoid this difficulty is to introduce what are called_____.

Answer 25

Translating an object on a screen does not correspond directly to matrix multiplication because translation is not a linear transformation. The standard way to avoid this difficulty is to introduce what are called homogeneous coordinates.

Question 26

In general, (X, Y, Z, H) are homogeneous coordinates for (x, y, z) if H != 0 and

Answer 26

In general, (X, Y, Z, H) are homogeneous coordinates for (x, y, z) if H != 0 and

x = X/H, y = Y/H, z = Z/H

Question 27

A subspace of Rn is any set H in Rn that has three properties:

Answer 27

1. The zero vector is in H
2. For each u and v in H, the sum u + v is in H.
3. For each u in H and each scalar c, the vector cu is in H

Question 28

The column space of a matrix A is _____

Answer 28

The column space of a matrix A is the set Col A of all linear combinations of the columns of A

Question 29

The null space of a matrix A is _____

Answer 29

The null space of a matrix A is the set Nul A of all solution of the homogeneous equation Ax = 0

Question 30

____ of m homogeneous linear equations in n unknowns is a subspace of Rn.

Answer 30

The set of alle solutions of a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of Rn.

Question 31

A basis for a subspace H of Rn is _____.

Answer 31

A basis for a subspace H of Rn is a linearly independent set in H that spans H.

Question 32

The set ___ is called the standard basis for Rn.

Answer 32

The set {e1, …, en} is called the standard basis for Rn.

Question 33

_____ of a matrix A form a basis for the column space of A.

Answer 33

The pivot columns of a matrix A form a basis for the column space of A.

Question 34

Suppose the set B = {b1, …, bp} is a basis for a subspace H. For each x in H, ______ are the weights c1, …, cp such that x = c1b1 + … + cpbp, and the vector in Rp

[x]_B = (c1, …, cp)

is called the coordinate vector of x (relative to B) or the B-coordinate vector of x.

Answer 34

Suppose the set B = {b1, …, bp} is a basis for a subspace H. For each x in H, the coordinates of x relative to the basis B are the weights c1, …, cp such that x = c1b1 + … + cpbp, and the vector in Rp

[x]_B = (c1, …, cp)

is called the coordinate vector of x (relative to B) or the B-coordinate vector of x.

Question 35

The correspondence x |-> [x]_B is a ____ between H and R2 that preserves _____. We call such a correspondance an ____, and we that H is isomorphic to R2.

Answer 35

The correspondence x |-> [x]_B is a one-to-one correspondence between H and R2 that preserves linear combinations. We call such a correspondance an isomorphism, and we that H is isomorphic to R2.

Question 36

The dimension of a nonzero subspace H, denoted by dim H, is ____.

Answer 36

The dimension of a nonzero subspace H, denoted by dim H, is the number of vector in any bassi for H. The dimension of the zero subspace {0} is defined to be zero.

Question 37

The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.

Answer 37

The rank of a matrix A, denoted by rank A, is ___.

Question 38

If a matrix A has n columns, then ___ + ___ = n

Answer 38

If a matrix A has n columns, then rank A + dim Nul A = n

Question 39

Let H be a p-dimensional subspace of Rn. Any linearly independent set of exactly p elements in H is automatically ___. Also, any set of p elements of H that spand H is automatically ___.

Answer 39

Let H be a p-dimensional subspace of Rn. Any linearly independent set of exactly p elements in H is automatically a basis for H. Also, any set of p elements of H that spand H is automatically a basis for H.

Question 40

The Invertible Matrix Theorem (continued)

Answer 40

Let A be an n x n matrix. Then the following statements are each equivalent to the statment that A is an invertible matrix:

• The columns of A form a basis of Rn
• Col A = Rn
• dim Col A = n
• rank A = n
• Nul A = {0}
• dim Nul A = 0