Direct sum and is sum whole space

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Created this as a copy of WBQ 4.14 Direct sum and is sum whole space.

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For each of the following subspaces, determine whether the sum $U + W$ is a direct sum, and whether the sum gives the whole vector space $V$ (i.e. whether $U + W = V$ ). The parts will talk you through some of the steps.

Give any introductory information the student needs.

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a) Let $ U = \left\{ \begin{pmatrix} x\\x \end{pmatrix} \middle| x \in \mathbb{R} \right\} $ and $ W = \left\{ \begin{pmatrix} -y\\y \end{pmatrix} \middle| y \in \mathbb{R} \right\} $ be subspaces of $\mathbb{R}^2$.

We can see that if $v\in U\cap W$, then we have to have $v=\begin{pmatrix} x\\x \end{pmatrix}=\begin{pmatrix} -y\\y \end{pmatrix}$ for some $x$ and $y$. This means we need $x=-y$ and $x=y$, so the only option is $x=y=0$. So $v=0$ is the only vector in the intersection. So this is a direct sum.

We can get any general vector $\begin{pmatrix} a\\b\end{pmatrix} \in \mathbb{R}^2$ in this sum: $\begin{pmatrix} a\\b\end{pmatrix}=\begin{pmatrix} \frac{b+a}{2}\\\frac{b+a}{2}\end{pmatrix} +\begin{pmatrix} -\frac{b-a}{2}\\\frac{b-a}{2}\end{pmatrix}$. I.e. $x=\frac{b+a}{2}$ and $y=\frac{b-a}{2}$. There is only one option here.

b) Let $ U = \left\{ \begin{pmatrix} x\\x\\-x \end{pmatrix} \middle| x \in \mathbb{R} \right\} $ and $ W = \left\{ \begin{pmatrix} y\\z\\-z \end{pmatrix} \middle| y,z \in \mathbb{R} \right\} $ be subspaces of $ \mathbb{R}^3 $.

Here we can see that there is some overlap: for example, the vector $\begin{pmatrix}1\\1\\-1\end{pmatrix}$ is in $U\cap W$. In fact, $U\subseteq W$: any vector in $U$ is also in $W$. For any given $x$, we just have to choose $y=x$ and $z=x$. So this is not a direct sum.

It also does not give the whole space, as the second and third entry are very linked. So we cannot get the vector $\begin{pmatrix}0\\1\\0\end{pmatrix}$ or the vector $\begin{pmatrix}0\\0\\1\end{pmatrix}$ in the sum. (Or any vector where the third entry is different to minus the second entry.) In fact, since $U\subseteq W$, we get $U+W=W$.

c) Let $ U = \left\{ \begin{pmatrix} x\\y\\-y \end{pmatrix} \middle| x, y \in \mathbb{R} \right\} $ and $ W = \left\{ \begin{pmatrix} 0\\z\\-z \end{pmatrix} \middle| z \in \mathbb{R} \right\} $ be subspaces of $ \mathbb{R}^3 $.

Again we see some overlap: $ \begin{pmatrix} 0\\z\\-z \end{pmatrix} \in U\cap W$ for any $z$: we just put $x=0$ and $y=z$. So $W\subseteq U$. So it is not a direct sum.

Therefore $U+W=U$, which is not all of $\mathbb{R}^3$. As in b), we cannot get the vector $\begin{pmatrix}0\\1\\0\end{pmatrix}$ or the vector $\begin{pmatrix}0\\0\\1\end{pmatrix}$ in the sum.

d) Let $ P_2 $ be the vectorspace of polynomials of degree at most $ 2 $ and consider the subspaces $ U = \{ p=a_2 X^2 + a_1 X \vert a_1,a_2 \in \mathbb{R}\} $ and $ W = P_1= \{ q=b_1 X + b_0 \vert b_0,b_1 \in \mathbb{R}\}$.

There is an overlap here: the polynomial $X$ is in $U\cap W$, because $X=0\cdot X^2+1\cdot X=1\cdot X+0$. So of course any $dX\in U\cap W$. So this is not a direct sum. But we do not have one of the two spaces contained in the other: no polynomials in $U$ have any non-zero constant terms, and no polynomials in $W$ have any non-zero $X^2$ term.

However, we do get the whole space as the sum:

$c_2X^2+c_1X+c_0= (c_2X^2+c_1X) + (0X+c_0)$, or $c_2X^2+c_1X+c_0= (c_2X^2+0X) + (c_1X+c_0)$, or $c_2X^2+c_1X+c_0= (c_2X^2+0.4c_1X) + (0.6c_1X+c_0)$, or $c_2X^2+c_1X+c_0= (c_2X^2+(c_1+14)X) + ((c_1-14)X+c_0)$, etc. As long as $a_1+b_1=c_1$, and $a_2=c_2$, $b_0=c_0$, we can make this sum. So you see there is some redundancy coming from the overlap.

e) Let $ P_2 $ be the vectorspace of polynomials of degree at most $ 2 $ and consider the subspaces $ U = \{ p=a_2 X^2 \vert a_2 \in \mathbb{R}\} $ and $ W = P_0=\{ q=b_0 \vert b_0 \in \mathbb{R}\} $.

We see here that there is no overlap: if $p\in U\cap W$, then $p=a_2X^2 + 0X+0=0X^2+0X+b_0$, so $a_2=0$ and $b_0=0$, so $0$ is the only element in the intersection. So this is a direct sum.

The sum does not give all of $P_2$: we cannot get any non-zero $X$ term. So for example $X\notin U+W$, or any $c_2X^2+c_1X+c_0$ with $c_1\neq 0$.

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Julia Goedecke 2 years, 10 months ago

Julia Goedecke 2 years, 10 months ago

Julia Goedecke 2 years, 11 months ago

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