Linear Algebra Exam 1 concepts and theorems

Properties of the Transpose

(a) (A + B)T = AT + BT (b) (sA)T = sAT (c) (AT)T = A

properties of matrix addition and scalar multiplication

(a) A + B = B + A (b) (A + B) + C = A + (B + C) (c) A + O = A (d) A + (-A) = O (e) (st)A = s(tA) (f) s(A + B) = sA + sB (g) (s + t)A = sA + tA

Invertible Matrix Theorem

(a) A is invertible (b) The reduced row echelon form of A is In (c) The rank of A equals n (d) The span of columns A is Rn (e) The equation Ax = b is consistent for every b in Rn (f) The nullity of A is zero (g) The columns are linearly independent (h) The only solution of Ax = 0 is 0 (i) There exist an n x n matrix B such that BA = In (j) There exist an n x n matrix C such that AC = In (k) A is a product of elementary matrices

Properties of Matrix-Vector Products

(a) A(u+v) = Au +Av (b) A(cu) = c(Au) = (cA) u for every scalar c (c) (A + B)u = Au + Bu (d) Aej = aj for j = 1,2...,n (e) if B is an mxn matrix such that Bw = Aw for all w in Rn then B = A (f) A0 is the mx1 zero vector (g) if O is the mxn zero vector matrix then Ov is mx1 zero vector (h) Iv = v

Theorem 1.8

(a) The columns of A are linearly independent (b) The equation Ax=b has at most one solution for each b in Rm (c) the nullity of A is zero

Test for Consistancy

(a) The matrix equation Ax= b is consistent (b) the vector b is a linear combination of A (c) The rref of the augmented matrix [Ab] has no row of the form [0 0 ... 0 d] where d != 0

Theorem 2.2

(a) if A is invertible then A-1 is invertible and (A-1)-1 = A (b) if A and B are invertible then AB is invertible and (AB)-1 = B-1 * A-1 (c) if A is invertible, then AT is invertible (AT)-1 = (A-1)T

Theorem 2.1

(a) s(AC) = (sA)C (b) A(CP) = (AC)P (c) (A+B)C = AC + BC (d) C(P + Q) = CP +CQ (e) IkA = A = AIm (f) the product of any matrix and a zero matrix is a zero matrix (g) (AC)T = CT AT

Theorem 2.4

(a) the pivot columns of A are linearly independent (b) Each nonpivot column of A is a linear combination of the previous pivot columns of A, where the coefficients of the linear combinations are the entries of the corresponding column of the reduced row echelon form of A

Theorem 1.6

(a) the span of the columns of is Rm (b) The equation Ax=b has at least one solution (that ism Ax=b is consistent) for each b in Rm (c) The rank of A is m, the number of rows of A (d) the reduced row echelon form of A has no zero rows (e) there is a pivot position in each row of A.

Theorem 2.5

A is invertible if and only if the reduced row echelon form of A is Im

Theorem 2.3

Let A be an m x n matrix with reduced row echelon form R. Then there exist an invertible m x m matrix P such that PA = R

Theorem 1.7

Let S = {u1,u2..uk} be a set of vectors from Rn and let v be a vector in Rn then Span {u1,u2...uk,v} = span {u1,u2...uk} if and only if v belongs to the span of S

Theorem 1.9

Vectors u1,u2,...uk in Rn are linearly dependent if and only if u1 = 0 or there exist an i >= 2 such that ui is a linear combination of the preceding vectors u1,u2,...uk.