LINEAR ALGEBRA: OVERVIEW
HAROLD SULTAN
A BSTRACT. Review Sheet
1. D EFINITIONS The following are a list of definitions which you should be familiar with: (1) Linear combination (2) Consistent (3) Elementary row operations (4) row echelon from (5) Pivot column (6) Rank, Nullity of a Matrix (or linear transformation) (7) Span (8) Linearly (in)dependent (9) Domain, Range, Codomain L (10) linear transformation (11) Onto (12) One to one (13) Subspace (14) Basis (15) Dimension (16) Similar (17) Eigenvector (18) Eigenvalue (19) Characteristic equation (20) Diagonalizable (21) Norm (22) Orthogonal (23) Orthogonal matrix (24) Symmetric matrix 2. THEOREMS The following are a list of theorems which you should be comfortable with: Theorem 2.1. (Properties of matrix addition and scalar multiplication) Let A,B,C be mxn matrices and let let s, t ∈ R. Then (1) A+B=B+A (2) (A+B)+C=A +(B+C) (3) A+0n =A
Date: 8/10/08.
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HAROLD SULTAN
(4) (5) (6) (7) (8) (9)
A+(-A)=0n st(A)=s(tA) s(A+B)=sA+sB (A + B)T = AT + B T (SA)T = sAT (AT )T = A
Theorem 2.2. (Properties of matrix vector products) Let A,B be mxn matrices and let u, v ∈ Rn , c ∈ R. Then (1) Aej = aj where ej is the jth standard vector and aj is the jth column vector of A (2) A(u+v)=Au+Av (3) A(cu)=cAu (4) (A+B)u=Au+Bu → → − − (5) A 0 = 0 → − (6) 0n u = 0 (7) In u = u Theorem 2.3. (Properties of matrix multiplication) Let A,B be kxm matrices, C a mxn matrix, and P,Q nxp matrices. Then (1) s(AC)=(sA)C=A(sC) (2) C(P+Q)=CP+CQ (3) Ik A = A = AIm (4) (AC)T = C T AT Theorem 2.4. Let A,B be nxn matrices. Then (1) if A is invertible, then A−1 is invertible and (A−1 )−1 = A (2) if A and B are invertible, then AB is invertible and (AB)−1 = B −1 A−1 (3) if A is invertible, then AT is invertible and (AT )−1 = (A−1 )T Theorem 2.5. *** (Invertible matrix theorem) Let A be an nxn matrix. Then TFAE: (1) A is invertible (2) ∃ B such that BA = In (3) ∃ C such that AC = In (4) the columns of A span Rn (5) ∀b ∈ Rn (∀= ”for all”) the system Ax=b is consistent (6) The rank of A is n (7) the columns of A are linearly independent → − (8) the only solution to Ax = 0 is the zero vector (9) the nullity of A is zero (10) det(A) = 0 Theorem 2.6. (Properties of determinant) Let A,B be an invertible matrix then: (1) det(AB) = det(A) det(B) (2) det(AT ) = det(A) 1 (3) det(A−1 ) = det(A) [Note: If A of B is not invertible then the first two of the above equalities still hold trivially, i.e. both sides are zero in both cases] Theorem 2.7. (Subspaces) Let V be a subspace of Rn of dimension k. Then any two of the following conditions about a subset S of V imply that s is a basis for V:
LINEAR ALGEBRA: OVERVIEW
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(1) S is linearly independent (2) S is a spanning set for V (3) S contains exactly k vectors Theorem 2.8. (properties of dot products) Let u, v, w ∈ Rn and let c be a scalar. Then 2 (1) u · u = u → − (2) u · u = 0 ⇐⇒ u = 0 (3) u · v = v · u (4) u · (v + w) = u · v + u · w (5) (cu) · v = c(u · v) = u · (cv) (6) cu = |c| u Theorem 2.9. (Cauchy-Schwartz inequality) Let u, v ∈ Rn , then |u · v| ≤ u ∗ v Theorem 2.10. (defining properties of Orthogonal Matrices) Let Q be an nxn matrix then TFAE: (1) Q is orthogonal (2) Q is invertible and QT = Q−1 (3) Qu · Qv = u · v for all u, v ∈ Rn (4) Qu = u for all u ∈ Rn Theorem 2.11. (determinant of orthogonal matrices) Let P,Q be orthogonal nxn matrix then: (1) det(Q) = ±1 (2) PQ is an orthogonal matrix (3) Q−1 is an orthogonal matrix 3. COMPUTATIONS The following are a list of some of the standard types of linear algebra computations you should be comfortable performing. (1) solving systems of equations (2) row reducing a matrix to echelon form (3) finding the rank and nullity of a matrix (4) matrix multiplication (5) finding the inverse of an invertible matrix (6) Cramer’s rule (7) computing the determinant of a matrix (8) finding the characteristic equation of a matrix (9) computing the eigenvalues and eigenvectors of a matrix (10) diagonalizing a matrix (11) change of basis (12) dot products (13) Gram-Schmidt process
E-mail address: HaroldSultan@gmail.com