Linear Algebra & Applications

2201NSC

Week 8 - Vector Spaces

Problem 1

Establish whether the following are vector spaces

  1. The set of all polynomials of the form $ax^3 - ax + b$
  2. The set of all vectors $(x_1,x_2,x_3)^T$ such that $x_1+x_3=1$
  3. The set of all vectors $(x_1,x_2,x_3)^T$ such that $x_1=x_2+x_3$
  4. The set of all complex numbers of the form $a + 2i$.



Problem 1 (cont.)

  1. The set of all polynomials of the form $ax^3 - ax + b$









Problem 2

Determine whether the following sets of vectors are linearly independent

  1. $\xx_1=(1,-1,2)^T,\ \xx_2=(-2,3,1)^T,\ \xx_3=(-1,3,8)^T$
  2. $\xx_1=\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix},\ \xx_2=\begin{pmatrix}0 & 2 \\ 0 & 0\end{pmatrix},\ \xx_3=\begin{pmatrix}2 & 3 \\ 0 & 2\end{pmatrix}$
  3. $\xx_1=1,\ \xx_2=e^x+e^{-x},\ \xx_3=e^x-e^{-x}$
  4. $\xx_1=1-x^2,\ \xx_2=2x^2,\ \xx_3=3$



Problem 2 (cont.)

a. $\xx_1=(1,-1,2)^T,\ \xx_2=(-2,3,1)^T,\ \xx_3=(-1,3,8)^T$











Problem 3

Find a basis for the parts of Q1 that represent vector spaces.











Problem 4

State why none of the following can be a basis for the stated vector space.

  1. $\left\{(1,-1)^T,(-2,1)^T,(-1,3)^T\right\}$ as a basis for $\R^2$
  2. $\left\{1-x, x^2-1, x^2-x\right\}$ as a basis for polynomials of degree 2 or less.
  3. $\left\{x^2-x,1+x,x^2+1,x^2-1\right\}$ as a basis for polynomials of degree 2 or less.
  4. $\left\{(1,-1,2)^T,(-2,3,1)^T,(-1,3,8)^T\right\}$ as a basis for $\R^3$



Problem 4 (cont.)

a. $\left\{(1,-1)^T,(-2,1)^T,(-1,3)^T\right\}$ as a basis for $\R^2$











Problem 5

Amend the sets of vectors given in the previous question to turn them into bases for the stated vector space.










Problem 6

Rewrite the following vector in terms of the basis provided

  1. $(1,2)^T$ using the basis $\left\{(1,-1)^T,(-2,1)^T\right\}.$
  2. $(1,0,0)^T$ using the basis \[\left\{(1,-1,2)^T,(-2,3,1)^T,(-1,3,1)^T\right\}.\]
  3. $\sinh 2x$ using the basis $\left\{e^{-2x}, e^{-x}, 1, e^{x}, e^{2x}\right\}$.
  4. $x^2$ using the basis $\left\{1-x, x^2-2, x^2-x\right\}.$



Problem 6 (cont.)

  1. $(1,2)^T$ using the basis $\left\{(1,-1)^T,(-2,1)^T\right\}.$










Problem 7

Give a basis and dimension for the column space, row space, and null space of the following matrices:

(a) $\,\begin{pmatrix}1 & 3 & 2 \\ 2 & 1 & 4 \\ 4 & 7 & 8\end{pmatrix}$ (c) $\,\begin{pmatrix}1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 4 \\ 0 & 2 & -1 & -1\end{pmatrix}$
(b) $\,\begin{pmatrix}1 & 3 & 2 \\ 2 & 1 & 4\end{pmatrix}$ (d) $\,\begin{pmatrix}1 & 1 & 0\\ 1 &0 & 1 \\ 0 & 0 & 1\end{pmatrix}$



Problem 7 (cont.)

(a) $\,\begin{pmatrix}1 & 3 & 2 \\ 2 & 1 & 4 \\ 4 & 7 & 8\end{pmatrix}$








Problems 8 & 9

8. Prove that if $\xx_1$ and $\xx_2$ are linearly independent vectors in $\R^4$ and $A\in\R^{4\times 4}$ is nonsingular, then $A\xx_1$ and $A\xx_2$ must also be linearly independent.


9. Prove that if $\v_1,\ldots,\v_k$ are vectors in vector space $V,$ and $ \text{Span}(\v_1,\ldots,\v_k) = \text{Span}(\v_1,\ldots,\v_{k-1}),$ then the vectors $\v_1,\ldots,\v_k$ are linearly dependent.