02 Matrix Algebra Flashcards
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 = ______
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]
Each column of AB is a __ of the columns of A using ____ from the corresponding column of B
Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B
rowi(AB) = __
rowi(AB) = rowi(A) B
The cancellation laws do ___ for matrix multiplication. That is, if AB = AC, then it is ___ in general that B = C.
The cancellation laws do not hold for matrix multiplication. That is, if AB = AC, then it is not true in general that B = C.
If a product AB is the zero matrix, you ___ conclude in general that either A = 0 or B = 0.
If a product AB is the zero matrix, you cannot conclude in general that either A = 0 or B = 0.
(A + B)T =
(A + B)T = AT + BT
(AB)T =
(AB)T = BTAT
The transpose of a product of matrices equals the product of their transposes _____
The transpose of a product of matrices equals the product of their transposes in the reverse order.
I (identity matrix) =
I = A-1A = AA-1
If ____, then A is invertible and
A-1 = ___
If ad - bc != 0, then A is invertible and
If A is an invertible n x n matrix, then for each b in Rn, the equation Ax = b has the ____ solution _____.
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-1b.
If A and B are n x n invertible matrices, then ____
If A and B are n x n invertible matrices, then so is AB
(AB)-1 =
(AB)-1 = B-1A-1
(AT)-1 =
(AT)-1 = (A-1)T
The product of n x n invertible matrices is invertible, and the inverse is the product of their inverses ___
The product of n x n invertible matrices is invertible, and the inverse is the product of their inverses in the reverse order
An elementary matrix is one that is obtained by ____.
An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix.
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 ____.
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 Im.
____ is the elementary matrix of the same type that transforms E back into I.
Each elementary matrix E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I.
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 ___.
An n x n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms Ininto A-1.
Algorithm for finding A-1
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.
The Invertible Matrix Theorem
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.
- AT is an invertible matrix.
Let A and B be square matrices. If AB = I, then A and B are ____
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.
invertible =
noninvertible =
invertible = nonsingular
noninvertible = singular
If the columns of a 4 x 3 matrix are linearly independent, ____ solutions to Ax = b.
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.
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_____.
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.
In general, (X, Y, Z, H) are homogeneous coordinates for (x, y, z) if H != 0 and
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
A subspace of Rn is any set H in Rn that has three properties:
- The zero vector is in H
- For each u and v in H, the sum u + v is in H.
- For each u in H and each scalar c, the vector cu is in H
The column space of a matrix A is _____
The column space of a matrix A is the set Col A of all linear combinations of the columns of A
The null space of a matrix A is _____
The null space of a matrix A is the set Nul A of all solution of the homogeneous equation Ax = 0
____ of m homogeneous linear equations in n unknowns is a subspace of Rn.
The set of alle solutions of a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of Rn.
A basis for a subspace H of Rn is _____.
A basis for a subspace H of Rn is a linearly independent set in H that spans H.
The set ___ is called the standard basis for Rn.
The set {e1, …, en} is called the standard basis for Rn.
_____ of a matrix A form a basis for the column space of A.
The pivot columns of a matrix A form a basis for the column space of A.
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.
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.
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.
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.
The dimension of a nonzero subspace H, denoted by dim H, is ____.
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.
The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.
The rank of a matrix A, denoted by rank A, is ___.
If a matrix A has n columns, then ___ + ___ = n
If a matrix A has n columns, then rank A + dim Nul A = n
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 ___.
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.
The Invertible Matrix Theorem (continued)
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
