Exam Revision

Question 1

Image space of A

Answer 1

The set of all vectors y ∈ R^m such that Ax = y has a solution

im(A) = {Ax : x ∈ R^n}.

Question 2

Null space of A

Answer 2

The subspace of the solutions to the homogenous equation Ax=0.

null(A) = {x ∈ R^n: Ax = 0}.

Question 3

Nullity of a matrix A

Answer 3

the dimension of its null space null(A), and is denoted nullity(A).

Question 4

The rank and nullity theorem

Answer 4

If A is an m × n matrix then
rank(A) + nullity(A) = n.

Question 5

Rank

Answer 5

dimension of row space

Question 6

Subspace of R^n

Answer 6

A set U of vectors in R^n such that:
(1) The zero vector 0 ∈ U.
(2) If x and y are in U then x+y is in U.
(3) If x is in U and c ∈ R is a scalar, then cx ∈ U.

Question 7

det(kA) =

Answer 7

k^n det(A)

Question 8

det(A^T) =

Answer 8

det(A)

Question 9

Rank of quadratic form

Answer 9

Let Q be a non-zero quadratic form in n variables that transforms such that f(x) = x^TAx, x ∈ Rn, A is a symmetric n × n matrix with p positive eigenvalues and n negative eigenvalues. Then p + n is the rank of f

Question 10

Signature of quadratic form

Answer 10

Let Q be a non-zero quadratic form in n variables that transforms such that f(x) = x^TAx, x ∈ Rn, A is a symmetric n × n matrix with p positive eigenvalues and n negative eigenvalues. Then p − n is the signature of f

Question 11

Unitary matrix

Answer 11

U^−1 = U^H

Question 12

Theorem 5.5.1 Let A and B be n × n matrices with A ∼ B. Then:

Answer 12

(a) det(A) = det(B);
(b) A is invertible if and only if B is invertible;
(c) A and B have the same rank;
(d) A and B have the same characteristic polynomial;
(e) A and B have the same eigenvalues.
(f) A and B have the same trace.