2.9 Dimension of a Subspace

Question 1

dimension

Answer 1

Let V be a subspace of Rn. Then the dimension of V is the number of elements in a basis of V

Question 2

rank

Answer 2

dim(C(A))

Question 3

nullity

Answer 3

dim(N(A))

Question 4

The Rank Nullity Theorem

Answer 4

Let A be an m x n matrix, then

dim(Nul A) + dim(Col A) = n

Question 5

The Basis Thm

Answer 5

Suppose V subset of Rn is a subspace of Rn and dimV=p

( 1 ) Any set of linearly independent vectors is a basis for V
( 2 ) Any set of p vectors that spans V is a basis for V

Question 6

how many basis in R2?

Answer 6

infinitely many

Question 7

Coordinates of v with respect to (or relative to) the basis B

Answer 7

Let B={v1, v2,….,vn} be a basis for Rn. If V is in Rn, then v can be written as
c1v1 + c2v2 + … + cnvn

then the coord is

[v]basis = c1
c2
…
cn basis

Question 8

change of basis matrix

Answer 8

C=[v1 v2 … vn]

Rn(basis)—>Rn

Question 9

is the change of basis invertible

Answer 9

yes bc IMT