Combining Subspaces

Decide if $\mathbb{R}^{2}$ is the direct sum of each $W_{1}$ and $W_{2}$.

• $W_{1}=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ 0\end{pmatrix}\mid x\in\mathbb{R}\hskip 1.00006pt\}$}$, $W_{2}=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ x\end{pmatrix}\mid x\in\mathbb{R}\hskip 1.00006pt\}$}$

• $W_{1}=\text{$\{\hskip 1.00006pt\begin{pmatrix}s\\ s\end{pmatrix}\mid s\in\mathbb{R}\hskip 1.00006pt\}$}$, $W_{2}=\text{$\{\hskip 1.00006pt\begin{pmatrix}s\\ 1.1s\end{pmatrix}\mid s\in\mathbb{R}\hskip 1.00006pt\}$}$

• $W_{1}=\mathbb{R}^{2}$, $W_{2}=\text{$\{\hskip 1.00006pt\vec{0}\hskip 1.00006pt\}$}$

• $W_{1}=W_{2}=\text{$\{\hskip 1.00006pt\begin{pmatrix}t\\ t\end{pmatrix}\mid t\in\mathbb{R}\hskip 1.00006pt\}$}$

• $W_{1}=\text{$\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\end{pmatrix}+\begin{pmatrix}x\\ 0\end{pmatrix}\mid x\in\mathbb{R}\hskip 1.00006pt\}$}$, $W_{2}=\text{$\{\hskip 1.00006pt\begin{pmatrix}-1\\ 0\end{pmatrix}+\begin{pmatrix}0\\ y\end{pmatrix}\mid y\in\mathbb{R}\hskip 1.00006pt\}$}$

Show that $\mathbb{R}^{3}$ is the direct sum of the $xy$-plane with each of these.

• the $z$-axis

• the line

  $\{\hskip 1.00006pt\begin{pmatrix}z\\ z\\ z\end{pmatrix}\mid z\in\mathbb{R}\hskip 1.00006pt\}$

Is $\mathcal{P}_{2}$ the direct sum of $\{\hskip 1.00006pta+bx^{2}\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$ and $\{\hskip 1.00006ptcx\mid c\in\mathbb{R}\hskip 1.00006pt\}$?

In $\mathcal{P}_{n}$, the even polynomials are the members of this set

and the odd polynomials are the members of this set.

Show that these are complementary subspaces.

Which of these subspaces of $\mathbb{R}^{3}$

can be combined to

• (a)  sum to $\mathbb{R}^{3}$?     (b)  direct sum to $\mathbb{R}^{3}$?

Show that $\mathcal{P}_{n}=\text{$\{\hskip 1.00006pta_{0}\mid a_{0}\in\mathbb{R}\hskip 1.00006pt\}$}\oplus\dots\oplus\text{$\{\hskip 1.00006pta_{n}x^{n}\mid a_{n}\in\mathbb{R}\hskip 1.00006pt\}$}$.

What is $W_{1}+W_{2}$ if $W_{1}\subseteq W_{2}$?

Does Example 4.5 generalize? That is, is this true or false: if a vector space $V$ has a basis $\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle$ then it is the direct sum of the spans of the one-dimensional subspaces $V=[\text{$\{\hskip 1.00006pt\vec{\beta}_{1}\hskip 1.00006pt\}$}]\oplus\dots\oplus[\text{$\{\hskip 1.00006pt\vec{\beta}_{n}\hskip 1.00006pt\}$}]$?

Can $\mathbb{R}^{4}$ be decomposed as a direct sum in two different ways? Can $\mathbb{R}^{1}$?

This exercise makes the notation of writing ‘$+$’ between sets more natural. Prove that, where $W_{1},\dots,W_{k}$ are subspaces of a vector space,

and so the sum of subspaces is the subspace of all sums.

(Refer to Example 4.19. This exercise shows that the requirement that pairwise intersections be trivial is genuinely stronger than the requirement only that the intersection of all of the subspaces be trivial.) Give a vector space and three subspaces $W_{1}$, $W_{2}$, and $W_{3}$ such that the space is the sum of the subspaces, the intersection of all three subspaces $W_{1}\cap W_{2}\cap W_{3}$ is trivial, but the pairwise intersections $W_{1}\cap W_{2}$,

Prove that if $V=W_{1}\oplus\dots\oplus W_{k}$ then $W_{i}\cap W_{j}$ is trivial whenever $i\neq j$. This shows that the first half of the proof of Lemma 4.15 extends to the case of more than two subspaces. (Example 4.19 shows that this implication does not reverse; the other half does not extend.)

Recall that no linearly independent set contains the zero vector. Can an independent set of subspaces contain the trivial subspace?

Does every subspace have a complement?

Let $W_{1},W_{2}$ be subspaces of a vector space.

• Assume that the set $S_{1}$ spans $W_{1}$, and that the set $S_{2}$ spans $W_{2}$. Can $S_{1}\cup S_{2}$ span $W_{1}+W_{2}$? Must it?

• Assume that $S_{1}$ is a linearly independent subset of $W_{1}$ and that $S_{2}$ is a linearly independent subset of $W_{2}$. Can $S_{1}\cup S_{2}$ be a linearly independent subset of $W_{1}+W_{2}$? Must it?

When we decompose a vector space as a direct sum, the dimensions of the subspaces add to the dimension of the space. The situation with a space that is given as the sum of its subspaces is not as simple. This exercise considers the two-subspace special case.

• For these subspaces of $\mathcal{M}_{2\!\times\!2}$ find $W_{1}\cap W_{2}$, $\dim(W_{1}\cap W_{2})$, $W_{1}+W_{2}$, and $\dim(W_{1}+W_{2})$.

• Suppose that $U$ and $W$ are subspaces of a vector space. Suppose that the sequence $\langle\vec{\beta}_{1},\dots,\vec{\beta}_{k}\rangle$ is a bas…

A matrix is symmetric if for each pair of indices $i$ and $j$, the $i,j$ entry equals the $j,i$ entry. A matrix is antisymmetric if each $i,j$ entry is the negative of the $j,i$ entry.

• Give a symmetric $2\!\times\!2$ matrix and an antisymmetric $2\!\times\!2$ matrix. (Remark. For the second one, be careful about the entries on the diagonal.)

• What is the relationship between a square symmetric matrix and its transpose? Between a square ant…

Let $W_{1},W_{2},W_{3}$ be subspaces of a vector space. Prove that $(W_{1}\cap W_{2})+(W_{1}\cap W_{3})\subseteq W_{1}\cap(W_{2}+W_{3})$. Does the inclusion reverse?

The example of the $x$-axis and the $y$-axis in $\mathbb{R}^{2}$ shows that $W_{1}\oplus W_{2}=V$ does not imply that $W_{1}\cup W_{2}=V$. Can $W_{1}\oplus W_{2}=V$ and $W_{1}\cup W_{2}=V$ happen?

We know that if $V=W_{1}\oplus W_{2}$ then there is a basis for $V$ that splits into a basis for $W_{1}$ and a basis for $W_{2}$. Can we make the stronger statement that every basis for $V$ splits into a basis for $W_{1}$ and a basis for $W_{2}$?

We can ask about the algebra of the ‘$+$’ operation.

• Is it commutative; is $W_{1}+W_{2}=W_{2}+W_{1}$?

• Is it associative; is $(W_{1}+W_{2})+W_{3}=W_{1}+(W_{2}+W_{3})$?

• Let $W$ be a subspace of some vector space. Show that $W+W=W$.

• Must there be an identity element, a subspace $I$ such that $I+W=W+I=W$ for all subspaces $W$?

• Does left-cancellation hold: if $W_{1}+W_{2}=W_{1}+W_{3}$ then $W_{2}=W_{3}$? Right cancellation?