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Math 511: Linear Algebra

Direct Sum

Direct Sum¶

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You will explore the sum and direct sum of vector spaces.

1. Consider the subspaces of $V = \mathbb{R}^3$ below,¶

$$ \begin{align*} U &= \left\{ \langle x,y,x-y\rangle\,:\,x,y\in\mathbb{R}\right\} \\ \\ W &= \left\{ \langle x,0,x\rangle\,:\,x\in\mathbb{R}\right\} \\ \\ Z &= \left\{ \langle x,x,x \rangle\,:\,x\in\mathbb{R}\right\} \end{align*} $$

Find $U + W$, $U + Z$, and $W + Z$.

2. Direct Sums¶

(a) Prove the following theorem,

Theorem¶

If $U$ and $W$ are subspaces of vector space $V$ such that $V = U + W$ and $U\cap W=\left\{\mathbf{0}\right\}$, then prove that every vector in $V$ has a unique representation of the form $\mathbf{u}+\mathbf{w}$, where $\mathbf{u}\in U$ and $\mathbf{w}\in W$. $V$ is called the direct sum of $U$ and $W$, and is written $V = U\oplus W$.

(b) Which of the sums in problem 1 are direct sums?

3. Prove the following theorem,¶

Theorem¶

Let vector space $V = U\oplus W$ be the direct sum of two subspaces $U$ and $W$, and let $\left\{ \mathbf{u}_1,\ \mathbf{u}_2,\dots,\mathbf{u}_k\right\}$ be a basis for $U$ and $\left\{\mathbf{w}_1,\ \mathbf{w}_2,\dots,\mathbf{w}_m\right\}$ be a basis for $W$. Prove that the set $S = \left\{\mathbf{u}_1,\ \mathbf{u}_2,\dots,\mathbf{u}_k,\ \mathbf{w}_1,\ \mathbf{w}_2,\dots,\mathbf{w}_m\right\}$ is a basis for $V$.

4. Consider the subspaces $U = \left\{\langle x,0,y\rangle\,:\,x,y,\in\mathbb{R}\right\}$ and $W = \left\{\langle 0,x,y \rangle\,:\,x,y\in\mathbb{R}\right\}$ of $V=\mathbb{R}^3$.¶

(a) Is $\mathbb{R}^3$ a direct sum of $U$ and $W$?

(b) What are the dimensions of $U$, $W$, $U\cap W$ and $U + W$?

(c) Formulate a conjecture that relates the dimensions of $U$, $W$, $U\cap W$, and $U+W$.

5. Do there exist two two-dimensional subspaces of $\mathbb{R}^3$ whose intersection is the zero vector? Why or why not?¶

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