Dimension

Find a basis for, and the dimension of, $\mathcal{P}_{2}$.

Find a basis for, and the dimension of, the solution set of this system.

Find a basis for, and the dimension of, each space.

• $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\in\mathbb{R}^{4}\mid x-w+z=0\hskip 1.00006pt\}$

• the set of $5\!\times\!5$ matrices whose only nonzero entries are on the diagonal (e.g., in entry $1,1$ and $2,2$, etc.)

• $\text{$\{\hskip 1.00006pta_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\mid\text{$a_{0}+a_{1}=0$ and $a_{2}-2a_{3}=0$}\hskip 1.00006pt\}$}\subseteq\mathcal{P}_{3}$

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

Find the dimension of the vector space of matrices

subject to each condition.

• $a,b,c,d\in\mathbb{R}$

• $a-b+2c=0$ and $d\in\mathbb{R}$

• $a+b+c=0$, $a+b-c=0$, and $d\in\mathbb{R}$

Find the dimension of this subspace of $\mathbb{R}^{2}$.

Find the dimension of each.

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

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

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

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

What is the dimension of the span of the set $\{\hskip 1.00006pt\cos^{2}\theta,\sin^{2}\theta,\cos 2\theta,\sin 2\theta\hskip 1.00006pt\}$? This span is a subspace of the space of all real-valued functions of one real variable.

Find the dimension of $\mathbb{C}^{47}$, the vector space of $47$-tuples of complex numbers.

What is the dimension of the vector space $\mathcal{M}_{3\!\times\!5}$ of $3\!\times\!5$ matrices?

Show that this is a basis for $\mathbb{R}^{4}$.

(We can use the results of this subsection to simplify this job.)

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

• (a)  $\{\hskip 1.00006pt1,x^{2},x^{2}-x\hskip 1.00006pt\}$     (b)  $\{\hskip 1.00006ptx^{2}+x,x^{2}-x\hskip 1.00006pt\}$     (c)  $\{\hskip 1.00006pt2x^{2}+x+1,2x+1,2\hskip 1.00006pt\}$     (d)  $\{\hskip 1.00006pt3x^{2},-1,3x,x^{2}-x\hskip 1.00006pt\}$

Refer to Example 2.11.

• Sketch a similar subspace diagram for $\mathcal{P}_{2}$.

• Sketch one for $\mathcal{M}_{2\!\times\!2}$.

Where $S$ is a set, the functions $f\colon S\to\mathbb{R}$ form a vector space under the natural operations: the sum $f+g$ is the function given by $f+g\,(s)=f(s)+g(s)$ and the scalar product is $r\cdot f\,(s)=r\cdot f(s)$. What is the dimension of the space resulting for each domain?

• (a)  $S=\text{$\{\hskip 1.00006pt1\hskip 1.00006pt\}$}$     (b)  $S=\text{$\{\hskip 1.00006pt1,2\hskip 1.00006pt\}$}$     (c)  $S=\text{$\{\hskip 1.00006pt1,\ldots,n\hskip 1.00006pt\}$}$

(See Exercise 14.) Prove that this is an infinite-dimensional space: the set of all functions $f\colon\mathbb{R}\to\mathbb{R}$ under the natural operations.

(See Exercise 14.) What is the dimension of the vector space of functions $f\colon S\to\mathbb{R}$, under the natural operations, where the domain $S$ is the empty set?

Show that any set of four vectors in $\mathbb{R}^{2}$ is linearly dependent.

Show that $\langle\vec{\alpha}_{1},\vec{\alpha}_{2},\vec{\alpha}_{3}\rangle\subset\mathbb{R}^{3}$ is a basis if and only if there is no plane through the origin containing all three vectors.

Prove that any subspace of a finite dimensional space is finite dimensional.

Where is the finiteness of $B$ used in Theorem 2.4?

Prove that if $U$ and $W$ are both three-dimensional subspaces of $\mathbb{R}^{5}$ then $U\cap W$ is non-trivial. Generalize.

A basis for a space consists of elements of that space. So we are naturally led to how the property ‘is a basis’ interacts with operations $\subseteq$ and $\cap$ and $\cup$. (Of course, a basis is actually a sequence that it is ordered, but there is a natural extension of these operations.)

• Consider first how bases might be related by $\subseteq$. Assume that $U,W$ are subspaces of some vector space and that $U\subseteq W$. Can there exist b…

Consider how ‘dimension’ interacts with ‘subset’. Assume $U$ and $W$ are both subspaces of some vector space, and that $U\subseteq W$.

• Prove that $\dim(U)\leq\dim(W)$.

• Prove that equality of dimension holds if and only if $U=W$.

• Show that the prior item does not hold if they are infinite-dimensional.

Here is an alternative proof of this section’s main result, Theorem 2.4. First is an example, then a lemma, then the theorem.

• Express this vector from $\mathbb{R}^{3}$ a as a linear combination of members of the basis.

• In that combination pick a basis vector with a non-zero coefficient. Alter $B$ by exchanging $\vec{v}$ for that basis vector, to get a new sequence $\hat{B}$. Check that $\hat{B}$ is also a basis for $\mathbb{R}^{3}$.

• (Exchange Le…

A library has $n$ books and $n+1$ subscribers. Each subscriber read at least one book from the library. Prove that there must exist two disjoint sets of subscribers who read exactly the same books (that is, the union of the books read by the subscribers in each set is the same).

For any vector $\vec{v}$ in $\mathbb{R}^{n}$ and any permutation $\sigma$ of the numbers $1$, $2$, …, $n$ (that is, $\sigma$ is a rearrangement of those numbers into a new order), define $\sigma(\vec{v})$ to be the vector whose components are $v_{\sigma(1)}$, $v_{\sigma(2)}$, …, and $v_{\sigma(n)}$ (where $\sigma(1)$ is the first number in the rearrangement, etc.). Now fix $\vec{v}$ and let $V$ be the span of $\{\hskip 1.00006pt\sigma(\vec{v})\mid\text{$\sigma$ permutes $1$, \ldots, $n$}\hskip 1.00006pt\}$. What are the possibilities for the dimension of $V$?