2.8 Subspaces

Question 1

subspace

Answer 1

following three have to be satisfied:

1. zero vector belongs to S1
2. for all vectors x1 and x2 that belong to S, vector x1 + x2 belongs to S
3. for all vectors x belong to S, for all c that belong to R, cx belongs to S

Question 2

null space

Answer 2

the set of solutions to the homogeneous eqn Ax = 0

N(A) = {x belongs to Rn | Ax = 0}

Question 3

the null space of an m x n matrix is a __________

Answer 3

subspace of Rn

Question 4

column space

Answer 4

is the span of the columns of A

C(A) = span{a1 a2…an} subset of Rn

Question 5

basis

Answer 5

minimal set

Any set B belongs to V satisfying:

1. B is linearly independent
2. span B = V

Question 6

standard basis

Answer 6

the set B = {e1, e2, … , en} subset of Rn is a basis for Rn