Chapter 4 & 5 Theorems

Theorem 4 [4.3]

An indexed set {*v*1,...,*v*p} of two or more vectors with *v*1 != *0*, is LD iff some *v*j (with j > 1) is a linear combo of the preceding vectors, *v*1,...,*v*j-1.

Theorem 2 [5.1]

If *v*1,...,*v*r are eigenvectors that correspond to distinct eigenvalues λ1,...,λr of an n x n matrix A, then the set {*v*1,...*v*r} is LI.

Theorem 1 [4.1]

If *v*1,....,*v*p are in a vector space V, then Span{*v*1,....,*v*p} is a subspace of V.

Theorem 9 [4.5]

If a vector space V has a basis B = {*b*1,...*b*n}, then any set in V containing more than n vectors must be LD.

Theorem 10 [4.5]

If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors.

Theorem 4 [5.2]

If n x n matrices A and B are similar, then they have the same characteristic polys and hence the same eigenvalues (with the same multiplicities)

Theorem 13 [4.6]

If two matrices A and B are row equivalent, then their row spaces are the same. If B is in echelon form, the non-zero rows of B form a basis for the row space of A as well for that of B.

Theorem 3 (Properties of Determinants) [5.2]

Let A and B be n x n matrices. a. A is invertible iff det A != 0. b. det AB = (det A )(det B ) c. det Aᵀ = det A d. If A is triangular, then det A is the product of the entries on the main diagonal of A. e. A row replacement operation on A does not change the determinant. A row interchange changes the sign of the determinant. A row scaling also scales the determinant by the same scalar factor.

IMT (continued) [4.6]

Let A be an n x n matrix. TFAE: m. The columns of A form a basis for Rn. n. Col A = Rn o. dim Col A = n p. rank A = n q. Nul A = {*0*} r. dim Nul A = 0

IMT (continued) [5.2]

Let A be an n x n matrix. Then A is invertible iff: s. The number 0 is not an eigenvalue of A. t. The determinant of A is not zero.

Theorem 7 (Unique Represenation Theorem) [4.4]

Let B = {*b*1,...,*b*n} be a basis for a vector space V. Then for each *x* in V, there exists a unique set of scalars c1,....,cn such that *x* = c1*b*1 + ... + cn*b*n

Theorem 8 [4.4]

Let B = {*b*1,...,*b*n} be a basis for a vector space V. Then the coordinate mapping *x* |--> [*x*]ᵦ is a one-to-one linear transformation from V onto Rn.

Theorem 11 [4.5]

Let H be a subspace of a finite-dimensional vector space V. Any LI set in H can be expanded, if necessary, to a basis for H. Also, H is finite-dimensional and dim H <= dim V

Theorem 5 (Spanning Set Theorem) [4.3]

Let S = {*v*1,...,*v*p} be a set in V, and let H = Span{*v*1,...,*v*p}. a. If one of the vectors in S - say, *v*k - is a linear combo of the remaining vectors in S, then the set formed from S by removing *v*k still spans H. b. If H != {*0*}, some subset of S is a basis for H.

Theorem 12 (Basis Theorem) [4.5]

Let V be a p-dimensional vector space, p >= 1. Any LI set of exactly p elements in V is automatically a basis for V. Any set of exactly p elements that spans V is automatically a basis for V.

Theorem 3 [4.2]

The column space of an m x n matrix A is a subspace of Rm.

Theorem 14 (Rank Theorem) [4.6]

The dimensions of the column space and row space of an m x n matrix A are equal. This common dimension, the rank of A, also equals the number of pivot positions in A and satisfies the equation rank A + dim Nul A = n

Theorem 1 [5.1]

The eigenvalues of a triangular matrix are the entries on its main diagonal.

Theorem 2 [4.2]

The null space of an m x n matrix A is a subspace of Rn. Equivalently, the set of all solutions to a system A*x* = *0* of m homogenous linear equations in n unknowns is a subspace of Rn.

Theorem 6 [4.3]

The pivot columns of a matrix A form a basis for Col A.