Math 251 - Alexiades
Review for Exam 1
know and understand concepts / definitions / main results
Concepts
matrix operations, elementary row operations, REF, RREF, rank
transpose, diagonal, triangular, symmetric matrices
invertible matrix; Fundamental Thm for Square Matrices
determinant, cofactors, adjoint of A
vectors in ℝn: operations, norm, unit vectors, dot product
parallel vectors, orthogonal vectors
Precise definition of
elementary matrix, AT, equal matrices, row-equivalent matrices
symmetric matrix ( AT=A )
invertible n×n matrix A ( ∃ n×n matrix A-1 such that A-1A = AA-1 = I )
det(A) ( = sum of all signed elementary products from A )
eigenvalue of A ( is a scalar λ such that Ax = λx for some non-zero vector x )
eigenvector of A ( is a non-zero vector x such that Ax = λx for some scalar λ )
norm axioms
orthogonal vectors (u ⊥ v means u · v = 0 )
Methods: Know How To
write a linear m×n system in matrix form: Ax=b
Gauss elimination: reduce m×n matrix A to row-echelon form; find rank
Gauss-Jordan elimination: reduce m×n A to reduced row-echelon form; find rank
decide consistency of m×n system Ax = b ; solve Ax = b by Gauss elimination and by LU factorization of A
find A-1 by Gauss elimination (on [ A | I ]), and by LU factorization of A
find determinant by cofactor expansion
find eigenvalues and eigenvectors
vector operations, find length and direction (unit vector) of a vector
Basic / crucial results to know
properties of transpose, of inverse, of determinant
Cauchy-Schwarz Inequality (Thm 3.2.4)
Fundamental Theorem for square matrices:
(8 equivalent statements so far)
For a square n×n matrix A, the following are equivalent:
1. A is invertible
2. A ∼ In (i.e. A is row equivalent to the Identity)
3. A is expressible as a product of elementary matrices
4. rank(A) = n
5. the homogeneous system Ax = 0 has only the trivial solution
6. the nonhomogeneous system Ax = b has unique solution for any vector b ∈ Rn
7. det(A) ≠ 0
8. λ = 0 is not an eigenvalue of A