Basis and Dimension

Which of the following sets are a basis for $\mathbb{R}^3$ ?

For which value(s) of c do $\begin{bmatrix}1\\1\\0\end{bmatrix}, \begin{bmatrix}2\\c\\1\end{bmatrix}, \begin{bmatrix}c\\4\\3\end{bmatrix}$ form a basis for $\mathbb{R}^3$?

Consider the space of polynomials of degree 3 or lower, $\mathcal{P}_3$ . Give an example of a linearly independent set of vectors whose span is the entirety of $\mathcal{P}_3$ .

Let $A=\left[\begin{array}{rrr} 1&-1&1\\ \end{array}\right]$ and consider the subspace of $\mathbb{R}^3$:

$N=\{\overrightarrow{x}=(x,y,z)\in\mathbb{R}^3\;|\;A.\overrightarrow{x}=\overrightarrow{0}\}$

Show that $B=\{(1,1,0), (-1,0,1)\}$ is a basis for $N$

Consider the vector space $M_{2\times2}$ with the standard component-wise operations of addition and scalar multiplication

The subset $D=\left\{ \left[\begin{array}{cc} a&b\\ c&d \end{array}\right]\in M_{2\times2}\;\Big|\;a+d=b+c \right\}$ is a subspace of $M_{2\times2}$

Find a basis for $D$ and the dimension of $D$

Example: Verifying a Basis

Let $V$ be the vector space $\{(x,y)\;|\;x,y\in\mathbb{R}^+\}$ with operations:

• $(x_1,y_1)+(x_2,y_2)=(x_1x_2,\ y_1y_2)$

Suppose the set $B=\{\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\}$ is a basis for a vector space $V$

Define the vectors:

Find a basis for the following subspace of $\mathbb{R}^3$ and state its dimension:

$S:=\{\vec x\in\mathbb{R}^3\text{ | }\vec x\cdot\vec v=0\text{ for every } \vec v\in span\{(1,-1,1),(1,1,0)\}\}$

Determine whether the set $T=\{ (1,0,0),(0,-2,2), (1,1,1) \}$$$is a basis of $\mathbb{R}^3$.

In the real vector space $\mathbb{R}^3$ with the standard component-wise operations of vector addition and scalar multiplication, consider the subspace:

$W=\{(x,y,z)\in\mathbb{R}^3\;|\;y+3z=0\}$

Find a basis for $W$, and use your answer to state the dimension of $W$

Practice: Finding a Basis

Consider the vector space $M_{2\times2}$ with the standard component-wise operations of addition and scalar multiplication

The subset $D=\left\{ \left[\begin{array}{cc} a&b\\ c&d \end{array}\right]\in M_{2\times2}\;\Big|\;a+d=b+c \right\}$ is a subspace of $M_{2\times2}$

Example:

Show that for $\vec x,\vec y\in \mathbb{R}^2$ , if $Span\{\vec x,\vec y\}=\mathbb{R}^2$, then $$$\vec x,\vec y$ are linearly independent.

If vectors $\vu$, $\vv$, $\vx$ and $\vec{y}$ are vectors in $\mathbb{R}^7$ such that $\vu = -2\vv + 0\vx +4\vec{y}$, then a basis for $\text{span}\big\{ \vu,\, \vv,\, \vx,\, \vec{y} \big\}$ is $\big\{ \vu,\, \vv,\, \vec{y} \big\}$.

Let $\{ e_1, \dots, e_n \}$ be a basis for an $n$ - dimensional vector space $V$ , and let $L : V\rightarrow W$ be a linear map, where $$$W$ is also a vector space. What conditions are necessary for $\{ L(e_1), \dots, L(e_n) \}$ to be a basis for $W$ ?

Which of the following are a basis for the vector space of polynomials with degree at most 3?

Practice: Coordinates

Let $B_1= \left\{ \begin{bmatrix} -1\\ 2 \\0\\ \end{bmatrix} , \begin{bmatrix} 3\\ -5\\ 2\\ \end{bmatrix} , \begin{bmatrix} 4\\ -7\\ 3\\ \end{bmatrix} \right\}$ and $B_2= \left\{ \begin{bmatrix} 1\\ -4 \\3\\ \end{bmatrix} , \begin{bmatrix} 5\\ 2\\ -2\\ \end{bmatrix} , \begin{bmatrix} 4\\ -7\\ 0\\ \end{bmatrix} \right\}$ be two bases of $\mathbb{R}^3$.

If $[\vec x]_{\small B_1} = \begin{bmatrix} -4\\ 8\\ -7\\ \end{bmatrix}$, find:

Find a basis for the set of vectors $\bcb{\left\{ \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix},\, \begin{bmatrix} 2 \\ -2 \\ 0 \end{bmatrix},\, \begin{bmatrix} 0 \\ -2 \\ 2 \end{bmatrix}, \, \begin{bmatrix} 3 \\ -4 \\ 1\end{bmatrix} \right\}}$, and determine the dimension of their span.

Are the vectors $\bcb{\begin{bmatrix} 1 \\ 3 \\ 1 \\ 0 \end{bmatrix}, \, \begin{bmatrix} 2 \\ -1 \\ 0 \\ 0 \end{bmatrix}, \, \begin{bmatrix} 0 \\ 2 \\ 1 \\ -1 \end{bmatrix} , \, \begin{bmatrix} 3 \\ 6 \\ -3 \\ -2 \end{bmatrix} }$ a basis for $\bcb{\mathbb{R}^4}$?

• Verify if the following vectors form a basis for $\mathbb{R}^3$:

If it is known that the set of vectors $\bcb{\left\{ \vec{x},\, \vec{y},\, \vec{z},\, \vec{w} \right\}}$ form a basis for $\bcb{\mathbb{R}^4}$, show that the set $\bcb{\left\{ \vec{x} + k\vec{z}, \vec{y},\, \vec{z},\, \vec{w} \right\}}$ is also a basis, for any scalar $\bcb{k}$.

In the vector space $P_2$, what is the dimension of the subspace that contains all polynomials of the form $p(x)=a+bx+cx^2$, such that $a=b+c$?

Consider the vector space $\mathcal{M}_{2, 2} (\mathbb{R})$ . What is the dimension of this vector space?

Which of the following is a basis for $\mathbb{R}^2$?

Which of the following is a basis for $\mathbb{R}^2$?

Let $V=\{(x,y)\;|\;x\in\mathbb{R},y\in\mathbb{R}^+\}$ be the vector space with operations:

• $(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1y_2)$
• $k\cdot(x,y)=(kx,y^k)$

Which of the following is a basis for $\mathbb{R}^2$?

It is known that the set $\{\vv_1,\, \vv_2,\, \dots \,,\, \vv_n\}$ is a basis for $\mathbb{R^n}$. If $\vw \in$$\mathbb{R^n}$, is the set $\{\vv_1,\, \vv_2,\, \dots \,,\, \vv_n,\, \vw\}$ linearly in/dependent? Briefly justify your answer (using mathematical notation is preferable to words, if possible).

Let $\vec v_1,\vec v_2,\vec v_3, \vec v_4$ be any four vectors in $\mathbb{R}^4$

Explain why the set $C=\{\vec v_1-\vec v_2,\vec v_2-\vec v_3,\vec v_3-\vec v_4,\vec v_4-\vec v_1\}$ cannot be a basis for $\mathbb{R}^4$

Practice Question: Bases

For each of the following subspaces, find the dimension of the subspace by giving a specific example of a basis:

$Q:=span\{(0,0,0,0), (1,1,0,0),(0,0,1,1),(1,1,1,1)\}\subseteq\mathbb{R}^4$

Show that for $\vec x,\vec y\in \mathbb{R}^2$ , if $Span\{\vec x,\vec y\}=\mathbb{R}^2$, then $$$\vec x,\vec y$ are linearly independent.