Ch. 2 and 3 of Linear Algebra

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each column of it is a linear combination of the columns of A using weights from the corresponding columns of B, [Ab1 ... Abn]

the matrix product AB (words and symbols)

associative, left-distributive, right-distributive, position of scalar irrelevant, identity

properties of matrix multiplication

the matrix whose columns are formed from the corresponding rows of A

the transpose of an m x n matrix A (denoted A^T) is

A

What is the transpose of the transpose of A?

A^T + B^T (the sum of the transposes)

What is (A + B)^T (the transpose of the sum)?

rA^T

(rA) ^T = ? (r is any scalar)

(B^T)(A^T) (the product of the transposes in reverse order)

(AB)^T (the transpose of the product)

a matrix that when multiplied on the left or the right by A produces the identity matrix, CA=AC=I

The inverse of a square matrix A is (definition in words and symbols)

there is an n x n matrix C satisfying CA = AC = I

A n x n matrix is invertible if (definition)

x = A⁻¹b

If A is an invertible n x n matrix, then for each b in Rn, Ax = b has what unique solution?

A⁻¹ is invertible and (A⁻¹)⁻¹ is A

If A is invertible... (Theorem 2.6a)

AB is invertible and (AB)⁻¹ = B⁻¹A⁻¹

If A and B are invertible matrices... (Theorem 2.6b)

A^T is invertible and (A^T)⁻¹ = (A⁻¹)^T

If A is invertible... (Theorem 2.6c)

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

An elementary matrix is

A is row equivalent to In. In this case, any sequence of elementary row operations that reduces A to In will also transform In into A⁻¹

An n x n matrix is invertible if and only if (relation to In)...

they from a linearly independent set and span Rn

By the Invertible Matrix Theorem, if A is an invertible n x n matrix, what can be known about A's columns?

it's one-to-one and maps Rn onto Rn

By the Invertible Matrix Theorem, if A is an invertible n x n matrix, what can be known about the linear transformation x→Ax

A is row equivalent to In, there is an n x n matrix C such that CA = In, there is an n x n matrix D such that AD = In

By the Invertible Matrix Theorem, if A is an invertible n x n matrix, what can be known about A's relationship to In?

Ax = 0 has only the trivial solution, Ax = b has at least one solution for each b in Rn

By the Invertible Matrix Theorem, if A is an invertible n x n matrix, what can be known about equations involving Ax?

A has n pivot positions, A^T is an invertible matrix

By the Invertible Matrix Theorem, if A is an invertible n x n matrix, what are two kinda random facts that are known?

there exists a function S: Rn → Rn such that S(T(x)) = T(S(x)) = x for all x in Rn

A linear transformation T : Rn → Rn is invertible if

an equation A = LU that expresses a matrix A as the product of L (a lower triangular matrix with 1s on the main diagonal) and U (an echelon form of A)

An LU factorization is

col1Arow1B + .... + colnArownB where each term is an m x p matrix called an outer product

If A is m x n and B is n x p, AB is

the entry in the ith row and jth column of the adjugate is the cofactor of A, Cji

Define the adjugate (or classical adjoint) of a matrix A

the number Cij, where Cij = (-1)^(i+j) det Aij

The (i,j)- cofactor of a matrix A is

I detA I

If A is a 2 x 2 matrix, the area of the parallelogram determined by A's columns is...

I detA I

If A is a 3 x 3 matrix, the volume of the parallelepiped determined by the columns of A is...

I detA I (area of S)

If T is a linear transformation determined by a 2 x 2 matrix A and S is an area, what's the area of T(S)?

I detA I (volume of S)

If T is a linear transformation determined by a 3 x 3 matrix A and S is a volume, what's the volume T(S)?

A⁻¹ = (1 / detA) (adjA)

Suppose A is an invertible n x n matrix. What's a formula for it's inverse?

xi = (detAi(b)) / detA

If A is an invertible matrix, the unique solution to Ax=b is the vector x whose entries are given by....

(detA)(detB)

Let A and B be n x n matrices. What is det(AB)?

detA

Let A be an n x n matrix. What is det(A^T)

not invertible

detA = 0, A is...

invertible

det A ≠ 0, A is

the product of the main diagonal entries of A

If A is a triangular matrix, detA =

it doesn't

How does row replacement (replacing one row with the sum of itself and a multiple of another row) change a determinant?

it multiplies it by -1

How does interchanging two rows change a determinant?

the determinant also gets multiplied by the same scalar

How does scaling (multiplying one row by a scalar) change a determinant?

(-1)^r (product of the pivots in U)

If A is invertible and has been reduced to an echelon form U using r row interchanges, what is detA?

a11detA11 - a12detA12 + ... + (-1)^(1+n)a1ndetA1n = ∑ from j=1 to n of (-1)^(1+j)a1jdetA1j

The determinant of an n x n matrix A is given by

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