linear algebra

Intro to Abstract Vector Spaces

August 13, 2020

Question 1

Show that R4[x] is a vector space. That is, show that the addition and scalar multiplication we defined satisfies all the properties of being a vector space.

Question 2

a) Is 2+4x ∈ span

( 1 + x, 1 − 3x

)

b) Is 2 + 8x + 11x2 ∈ span

( 1 + 4x, 1 + 8x + 6x2,−1 − 12x−x2

)

c) Is 1 −x− 8x2 ∈ span

( 1, 1 + x + 4x2,−x− 4x2

) Question 3 For the following, give an example if one exists, or state it is not possible. If it is not possible, explain why. a) A sequence of 4 vectors that span M2×3(R) b) A sequence of 3 Linearly Independent vectors in R3[x]. c) A sequence of 3 Linearly independent vectors in R2[x] that are not a basis. d) A sequence of 9 spanning vectors in M3×3(R) that are not a basis.

Question 4

1

Consider the following two bases for R2[x]:

B = {1, x, x2} C = {2 + x, 3 + x, x−x2}

a) Find PB−→C (the change of basis matrix from B to C) b) Find [8 − 12x + 36x2]B c) Find [8 − 12x + 36x2]C

Question 5 Consider the following two bases for M2×2(R):

B = { (

1 0 0 0

) ,

( 0 1 0 0

) ,

( 0 0 1 0

) ,

( 0 0 0 1

) }

C = { (

1 1 1 2

) ,

( 1 2 1 1

) ,

( 1 1 2 1

) ,

( 1 1 1 1

) }

a) Suppose m is a matrix and [m]C =

 

3 8 12 −2

 . Find what the matrix m is.

b) Find [m]B c) Find PC−→B and confirm that [m]B= PC−→B[m]C

2