Subspaces and Spanning Sets

Which of these subsets of the vector space of $2\!\times\!2$ matrices are subspaces under the inherited operations? For each one that is a subspace, parametrize its description. For each that is not, give a condition that fails.

• $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ 0&b\end{pmatrix}\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ 0&b\end{pmatrix}\mid a+b=0\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ 0&b\end{pmatrix}\mid a+b=5\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt\begin{pmatrix}a&c\\ 0&b\end{pmatrix}\mid a+b=0,c\in\mathbb{R}\hskip 1.00006pt\}$

Is this a subspace of $\mathcal{P}_{2}$: $\{\hskip 1.00006pta_{0}+a_{1}x+a_{2}x^{2}\mid a_{0}+2a_{1}+a_{2}=4\hskip 1.00006pt\}$? If it is then parametrize its description.

Is the vector in the span of the set?

Decide if the vector lies in the span of the set, inside of the space.

• $\begin{pmatrix}2\\ 0\\ 1\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\hskip 1.00006pt\}$, in $\mathbb{R}^{3}$

• $x-x^{3}$, $\{\hskip 1.00006ptx^{2},2x+x^{2},x+x^{3}\hskip 1.00006pt\}$, in $\mathcal{P}_{3}$

• $\begin{pmatrix}0&1\\ 4&2\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}1&0\\ 1&1\end{pmatrix},\begin{pmatrix}2&0\\ 2&3\end{pmatrix}\hskip 1.00006pt\}$, in $\mathcal{M}_{2\!\times\!2}$

A superhero is at the origin of a two dimensional plane. The superhero has two devices, a hoverboard that moves any distance in the direction $\binom{3}{1}$ and a magic carpet that moves any distance in the direction $\binom{1}{2}$.

• An evil villain is hiding out in the plane at the point $(-5,7)$. How many hoverboard units and magic carpet units does the superhero have to move to get to the villain?

• Is there anywhere in the p…

Which of these are members of the span $[\text{$\{\hskip 1.00006pt\cos^{2}x,\sin^{2}x\hskip 1.00006pt\}$}]$ in the vector space of real-valued functions of one real variable?

• (a)  $f(x)=1$     (b)  $f(x)=3+x^{2}$     (c)  $f(x)=\sin x$     (d)  $f(x)=\cos(2x)$

Which of these sets spans $\mathbb{R}^{3}$? That is, which of these sets has the property that any three-tall vector can be expressed as a suitable linear combination of the set’s elements?

• (a)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 2\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 3\end{pmatrix}\hskip 1.00006pt\}$     (b)  $\{\hskip 1.00006pt\begin{pmatrix}2\\ 0\\ 1\end{pmatrix},\begin{pmatrix}1\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\hskip 1.00006pt\}$     (c)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 1\\ 0\end{pmatrix},\begin{pmatrix}3\\ 0\\ 0\end{pmatrix}\hskip 1.00006pt\}$     (d)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\\ 1\end{pmatrix},\begin{pmatrix}3\\ 1\\ 0\end{pmatrix},\begin{pmatrix}-1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}2\\ 1\\ 5\end{pmatrix}\hskip 1.00006pt\}$     (e)  $\{\hskip 1.00006pt\begin{pmatrix}2\\ 1\\ 1\end{pmatrix},\begin{pmatrix}3\\ 0\\ 1\end{pmatrix},\begin{pmatrix}5\\ 1\\ 2\end{pmatrix},\begin{pmatrix}6\\ 0\\ 2\end{pmatrix}\hskip 1.00006pt\}$

Parametrize each subspace’s description. Then express each subspace as a span.

• The subset $\{\hskip 1.00006pt(\begin{matrix}a&b&c\end{matrix})\mid a-c=0\hskip 1.00006pt\}$ of the three-wide row vectors

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

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

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

  $\{\hskip 1.00006pt\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mid\text{$2a-c-d=0$ and $a+3b=0$}\hskip 1.00006pt\}$

• The subset $\{\hskip 1.00006pta+bx+cx^{3}\mid a-2b+c=0\hskip 1.00006pt\}$ of $\mathcal{P}_{3}$

• The subset of $\mathcal{P}_{2}$ of quadratic polynomials $p$ such that $p(7)=0$

Find a set to span the given subspace of the given space. (Hint. Parametrize each.)

• the $xz$-plane in $\mathbb{R}^{3}$

• $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mid 3x+2y+z=0\hskip 1.00006pt\}$ in $\mathbb{R}^{3}$

• $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\mid 2x+y+w=0\text{\ and\ }y+2z=0\hskip 1.00006pt\}$ in $\mathbb{R}^{4}$

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

• The set $\mathcal{P}_{4}$ in the space $\mathcal{P}_{4}$

• $\mathcal{M}_{2\!\times\!2}$ in $\mathcal{M}_{2\!\times\!2}$

Is $\mathbb{R}^{2}$ a subspace of $\mathbb{R}^{3}$?

Decide if each is a subspace of the vector space of real-valued functions of one real variable.

• The even functions $\{\hskip 1.00006pt\text{$f\colon\mathbb{R}\to\mathbb{R}$}\mid f(-x)=f(x)\text{for all }x\hskip 1.00006pt\}$. For example, two members of this set are $f_{1}(x)=x^{2}$ and $f_{2}(x)=\cos(x)$.

• The odd functions $\{\hskip 1.00006pt\text{$f\colon\mathbb{R}\to\mathbb{R}$}\mid f(-x)=-f(x)\text{ for all }x\hskip 1.00006pt\}$. Two members are $f_{3}(x)=x^{3}$ and $f_{4}(x)=\sin(x)$.

Example 2.16 says that for any vector $\vec{v}$ that is an element of a vector space $V$, the set $\{\hskip 1.00006ptr\cdot\vec{v}\mid r\in\mathbb{R}\hskip 1.00006pt\}$ is a subspace of $V$. (This is simply the span of the singleton set $\{\hskip 1.00006pt\vec{v}\hskip 1.00006pt\}$.) Must any such subspace be a proper subspace?

An example following the definition of a vector space shows that the solution set of a homogeneous linear system is a vector space. In the terminology of this subsection, it is a subspace of $\mathbb{R}^{n}$ where the system has $n$ variables. What about a non-homogeneous linear system; do its solutions form a subspace (under the inherited operations)?

Give an example of each or explain why it would be impossible to do so.

• A nonempty subset of $\mathcal{M}_{2\!\times\!2}$ that is not a subspace.

• A set of two vectors in $\mathbb{R}^{2}$ that does not span the space.

Example 2.19 shows that $\mathbb{R}^{3}$ has infinitely many subspaces. Does every nontrivial space have infinitely many subspaces?

Finish the proof of Lemma 2.9.

Show that each vector space has only one trivial subspace.

Show that for any subset $S$ of a vector space, the span of the span equals the span $[[S]]=[S]$. (Hint. Members of $[S]$ are linear combinations of members of $S$. Members of $[[S]]$ are linear combinations of linear combinations of members of $S$.)

All of the subspaces that we’ve seen in some way use zero in their description. For example, the subspace in Example 2.3 consists of all the vectors from $\mathbb{R}^{2}$ with a second component of zero. In contrast, the collection of vectors from $\mathbb{R}^{2}$ with a second component of one does not form a subspace (it is not closed under scalar multiplication). Another example is Example 2.2, where the condition on the ve…

We can give a justification for the convention that the sum of zero-many vectors equals the zero vector. Consider this sum of three vectors $\vec{v}_{1}+\vec{v}_{2}+\vec{v}_{3}$.

• What is the difference between this sum of three vectors and the sum of the first two of these three?

• What is the difference between the prior sum and the sum of just the first one vector?

• What should be the difference between the prior sum of one vec…

Is a space determined by its subspaces? That is, if two vector spaces have the same subspaces, must the two be equal?

• Give a set that is closed under scalar multiplication but not addition.

• Give a set closed under addition but not scalar multiplication.

• Give a set closed under neither.

Show that the span of a set of vectors does not depend on the order in which the vectors are listed in that set.

Which trivial subspace is the span of the empty set? Is it

or some other subspace?

Show that if a vector is in the span of a set then adding that vector to the set won’t make the span any bigger. Is that also ‘only if’?

Subspaces are subsets and so we naturally consider how ‘is a subspace of’ interacts with the usual set operations.

• If $A,B$ are subspaces of a vector space, must their intersection $A\cap B$ be a subspace? Always? Sometimes? Never?

• Must the union $A\cup B$ be a subspace?

• If $A$ is a subspace of some $V$, must its set complement $V-A$ be a subspace?

(Hint. Try some test subspaces from Example 2.19.)

Does the span of a set depend on the enclosing space? That is, if $W$ is a subspace of $V$ and $S$ is a subset of $W$ (and so also a subset of $V$), might the span of $S$ in $W$ differ from the span of $S$ in $V$?

Is the relation ‘is a subspace of’ transitive? That is, if $V$ is a subspace of $W$ and $W$ is a subspace of $X$, must $V$ be a subspace of $X$?

Because ‘span of’ is an operation on sets we naturally consider how it interacts with the usual set operations.

• If $S\subseteq T$ are subsets of a vector space, is $[S]\subseteq[T]$? Always? Sometimes? Never?

• If $S,T$ are subsets of a vector space, is $[S\cup T]=[S]\cup[T]$?

• If $S,T$ are subsets of a vector space, is $[S\cap T]=[S]\cap[T]$?

• Is the span of the complement equal to the complement of the span?

Find a structure that is closed under linear combinations, and yet is not a vector space.