Ch. 2 and 3 of Linear Algebra
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
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 = ? (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....
Let A and B be n x n matrices. What is det(AB)?
Let A be an n x n matrix. What is det(A^T)
detA = 0, A is...
det A ≠ 0, A is
the product of the main diagonal entries of A
If A is a triangular matrix, detA =
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