# Intro to Abstract Vector Spaces

## Abstract Vector Spaces

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.

Properties of Vector Spaces

Abstract Vector Spaces Question 2

1. a) Is 2+4x ∈ span

( 1 + x, 1− 3x)

1. b) Is 2 + 8x + 11×2 ∈ span

( 1 + 4x, 1 + 8x + 6×2,−1− 12x− x2)

1. c) Is 1− x− 8×2 ∈ span

( 1, 1 + x + 4×2,−x− 4×2)

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

Consider the following two bases for R2[x]: B = {1, x, x2} C = {2 + x, 3 + x, x− x2}

1. a) Find PB−→C (the change of basis matrix from B to C) b) Find [8− 12x + 36×2]B c) Find [8− 12x + 36×2]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) }

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

Invertible and Elementary Matrices

### Determinants

Question 1

Determine if the following matrices are invertible. (No need to find the inverse)

a)A= ( 2 3 4 5)

1. b) A=1 0 00 2 1 1 0 1.
1. c) A=1 0 12 1 3 3 0 3.

Determinants Question 2

Compute the determinants of the following matrices using any method you want.

1. a) A=1 0 −12 4 0 1 3 1.
2. b) B= 1 2 3 9 0 3 4 6 1 0 5 1 0 0 0 1.
3. c) C= 1 2 3 4 5 0 4 6 9 −26 0 0 8 12 18 2 0 0 −2 3 0 0 0 0 4.

Question 3

Say whether the following are true or false. If false explain why or give a counter example.

1. a) det(A+B)= det(A) + det(B) for any n× n matrices A,B.
2. b) Suppose A and B are two 36×36 matrices with det(A)=1063 and det(B)=2. Then the matrix AB is invertible.
3. c) Suppose A is a 3 × 3 matrix with det(A)=12. There exists some vec- tor ~b ∈ R3 such that there is no ~x ∈ R3 with A~x = ~b.
4. d) Suppose A is a 10, 340 × 10, 340 matrix with det(A)=0. Then there is some nonzero vector ~x ∈ R10,340 such that A~x = ~0