Basis

Decide if each is a basis for $\mathcal{P}_{2}$.

• (a)  $\langle x^{2}-x+1,2x+1,2x-1\rangle$     (b)  $\langle x+x^{2},x-x^{2}\rangle$

Decide if each is a basis for $\mathbb{R}^{3}$.

• (a)  $\langle\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},\begin{pmatrix}3\\ 2\\ 1\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\rangle$     (b)  $\langle\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},\begin{pmatrix}3\\ 2\\ 1\end{pmatrix}\rangle$     (c)  $\langle\begin{pmatrix}0\\ 2\\ -1\end{pmatrix},\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\begin{pmatrix}2\\ 5\\ 0\end{pmatrix}\rangle$     (d)  $\langle\begin{pmatrix}0\\ 2\\ -1\end{pmatrix},\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\begin{pmatrix}1\\ 3\\ 0\end{pmatrix}\rangle$

Represent the vector with respect to the basis.

• $\begin{pmatrix}1\\ 2\end{pmatrix}$, $B=\langle\begin{pmatrix}1\\ 1\end{pmatrix},\begin{pmatrix}-1\\ 1\end{pmatrix}\rangle\subseteq\mathbb{R}^{2}$

• $x^{2}+x^{3}$, $D=\langle 1,1+x,1+x+x^{2},1+x+x^{2}+x^{3}\rangle\subseteq\mathcal{P}_{3}$

• $\begin{pmatrix}0\\ -1\\ 0\\ 1\end{pmatrix}$, ${\cal E}_{4}\subseteq\mathbb{R}^{4}$

Represent the vector with respect to each of the two bases.

Find a basis for $\mathcal{P}_{2}$, the space of all quadratic polynomials. Must any such basis contain a polynomial of each degree: degree zero, degree one, and degree two?

Find a basis for the solution set of this system.

Find a basis for $\mathcal{M}_{2\!\times\!2}$, the space of $2\!\times\!2$ matrices.

Find a basis for each.

• The subspace $\{\hskip 1.00006pta_{2}x^{2}+a_{1}x+a_{0}\mid a_{2}-2a_{1}=a_{0}\hskip 1.00006pt\}$ of $\mathcal{P}_{2}$

• The space of three-wide row vectors whose first and second components add to zero

• This subspace of the $2\!\times\!2$ matrices

  $\{\hskip 1.00006pt\begin{pmatrix}a&b\\ 0&c\end{pmatrix}\mid c-2b=0\hskip 1.00006pt\}$

Find a basis for each space, and verify that it is a basis.

• The subspace $M=\text{$\{\hskip 1.00006pta+bx+cx^{2}+dx^{3}\mid a-2b+c-d=0\hskip 1.00006pt\}$}$ of $\mathcal{P}_{3}$.

• This subspace of $\mathcal{M}_{2\!\times\!2}$.

Check Example 1.6.

Find the span of each set and then find a basis for that span.

• (a)  $\{\hskip 1.00006pt1+x,1+2x\hskip 1.00006pt\}$ in $\mathcal{P}_{2}$     (b)  $\{\hskip 1.00006pt2-2x,3+4x^{2}\hskip 1.00006pt\}$ in $\mathcal{P}_{2}$

Find a basis for each of these subspaces of the space $\mathcal{P}_{3}$ of cubic polynomials.

• The subspace of cubic polynomials $p(x)$ such that $p(7)=0$

• The subspace of polynomials $p(x)$ such that $p(7)=0$ and $p(5)=0$

• The subspace of polynomials $p(x)$ such that $p(7)=0$, $p(5)=0$, and $p(3)=0$

• The space of polynomials $p(x)$ such that $p(7)=0$, $p(5)=0$, $p(3)=0$, and $p(1)=0$

We’ve seen that the result of reordering a basis can be another basis. Must it be?

Can a basis contain a zero vector?

Let $\langle\vec{\beta}_{1},\vec{\beta}_{2},\vec{\beta}_{3}\rangle$ be a basis for a vector space.

• Show that $\langle c_{1}\vec{\beta}_{1},c_{2}\vec{\beta}_{2},c_{3}\vec{\beta}_{3}\rangle$ is a basis when $c_{1},c_{2},c_{3}\neq 0$. What happens when at least one $c_{i}$ is $0$?

• Prove that $\langle\vec{\alpha}_{1},\vec{\alpha}_{2},\vec{\alpha}_{3}\rangle$ is a basis where $\vec{\alpha}_{i}=\vec{\beta}_{1}+\vec{\beta}_{i}$.

Find one vector $\vec{v}$ that will make each into a basis for the space.

• (a)  $\langle\begin{pmatrix}1\\ 1\end{pmatrix},\vec{v}\rangle$ in $\mathbb{R}^{2}$     (b)  $\langle\begin{pmatrix}1\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\vec{v}\rangle$ in $\mathbb{R}^{3}$     (c)  $\langle x,1+x^{2},\vec{v}\rangle$ in $\mathcal{P}_{2}$

Consider $2+4x^{2},1+3x^{2},1+5x^{2}\in\mathcal{P}_{2}$.

• Find a linear relationship among the three.

• Represent them with respect to $B=\langle 1-x,1+x,x^{2}\rangle$.

• Check that the same linear relationship holds among the representations, as in Lemma 1.18.

Where $\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle$ is a basis, show that in this equation

each of the $c_{i}$’s is zero. Generalize.

A basis contains some of the vectors from a vector space; can it contain them all?

Theorem 1.12 shows that, with respect to a basis, every linear combination is unique. If a subset is not a basis, can linear combinations be not unique? If so, must they be?

A square matrix is symmetric if for all indices $i$ and $j$, entry $i,j$ equals entry $j,i$.

• Find a basis for the vector space of symmetric $2\!\times\!2$ matrices.

• Find a basis for the space of symmetric $3\!\times\!3$ matrices.

• Find a basis for the space of symmetric $n\!\times\!n$ matrices.

We can show that every basis for $\mathbb{R}^{3}$ contains the same number of vectors.

• Show that no linearly independent subset of $\mathbb{R}^{3}$ contains more than three vectors.

• Show that no spanning subset of $\mathbb{R}^{3}$ contains fewer than three vectors. Hint: recall how to calculate the span of a set and show that this method cannot yield all of $\mathbb{R}^{3}$ when we apply it to fewer than three vectors.

One of the exercises in the Subspaces subsection shows that the set

is a vector space under these operations.

Find a basis.