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

Challenge Question for your enjoyment

Consider the matrix A=1 2 41 3 9 1 4 16. This matrix has det(A)=2=(4-3)(4- 2)(3-2) where I’ve expressed 2 in this weird way as a hint.

Now consider the matrix B= 1 2 4 8 1 3 9 27 1 4 16 64 1 5 25 125. This has det(B)=12=(5- 2 4)(5-3)(5-2)(4-2)(4-3) again written in this weird way as a hint.

Without plugging into a calculator, find the determinant of the matrix

C =1 2 4 8 16 1 3 9 27 81 1 4 16 64 256 1 5 25 125 625 1 6 36 216 1, 296 .

Such matrices are called ”Vandermonde Matrices” and they actually turn up in mathematics. For example, in so called ”algebraic number theory” (more specifically in finite number field extensions) in ”Galois theory” and in ”Group Representation Theory” this matrix is used in proving some key results. Furthermore, it is also used in Error correcting codes, and in com- puting Discrete Fourier Transforms, with applications to music.

Last Updated on August 16, 2020 by Essay Pro