# M216: Exercise sheet 2

## Warmup questions

• 1. Let $$U,W\leq V$$ be subspaces of a vector space $$V$$.

When is $$U\cup W$$ also a subspace of $$V$$?

• 2. Let $$V,W$$ be vector spaces, $$\lst {v}1n$$ a basis of $$V$$ and $$\lst {w}1n$$ a list of vectors in $$W$$. Let $$\phi :V\to W$$ be the unique linear map with

\begin{equation*} \phi (v_i)=w_i, \end{equation*}

for all $$1\leq i\leq n$$. Show:

• (a) $$\phi$$ injects if and only if $$\lst {w}1n$$ is linearly independent.

• (b) $$\phi$$ surjects if and only if $$\lst {w}1n$$ spans $$W$$.

Deduce that $$\phi$$ is an isomorphism if and only if $$\lst {w}1n$$ is a basis for $$W$$.

## Homework

• 3. Let $$V$$ be a vector space. A linear map $$\pi :V\to V$$ is called a projection if $$\pi \circ \pi =\pi$$.

In this case, prove that $$\ker \pi \cap \im \pi =\set {0}$$ and deduce that $$V=\ker \pi \oplus \im \pi$$.

• 4. Let $$U_1,U_2,U_3\leq \R ^3$$ be the $$1$$-dimensional subspaces spanned by $$(1,2,0)$$, $$(1,1,1)$$ and $$(2,3,1)$$ respectively.

Which of the following sums are direct?

• (a) $$U_i+U_j$$, for $$1\leq i<j\leq 3$$.

• (b) $$U_1+U_2+U_3$$.

• 5. Let $$V_1,V_2,V_3\leq V$$. Which of the following statements are true? (In each case, give a proof or a counter-example.)

• (a) $$V_1+(V_2\cap V_3)=(V_1+V_2)\cap (V_1+V_3)$$.

• (b) $$V_1\cap (V_2+V_3)=(V_1\cap V_2)+(V_1\cap V_3)$$.

• (c) $$(V_1\cap V_2)+(V_1\cap V_3)\subseteq V_1\cap (V_2+V_3)$$.

• 6. Let $$V_1,V_1',V_2\leq V$$ and suppose that $$V=V_1\oplus V_2$$ and $$V=V_1'\oplus V_2$$.

• (a) Must $$V_1=V_1'$$?

• (b) Are $$V_1$$ and $$V_1'$$ isomorphic?

Please hand in at 4W level 1 by NOON on Friday 20th October

# M216: Exercise sheet 2—Solutions

• 1. If $$U\subseteq W$$ then $$U\cup W=W$$ is a subspace and similarly if $$W\subseteq U$$. In any other case, $$U\cup W$$ is not a subspace: we can find $$u\in U\setminus W$$ and $$w\in W\setminus U$$ and then $$u+w\notin U$$ (else $$w=(u+w)-u\in U$$) and similarly $$u+w\notin W$$. Thus $$U\cup W$$ is not closed under addition.

• 2.

• (a) $$\lc {\lambda }w1n=0$$ if and only if $$\lc {\lambda }v1n\in \ker \phi$$. Thus $$\lst {w}1n$$ is linearly independent if and only if $$\phi$$ has trivial kernel.

• (b) $$\phi$$ surjects if and only if any $$w\in W$$ can be written $$w=\phi (v)$$, or equivalently,

\begin{equation*} w=\phi (\lc {\lambda }v1n)=\lc {\lambda }{w}1n, \end{equation*}

for some $$\lambda _i$$, $$1\leq i\leq n$$.

• 3. Let $$v\in \ker \pi \cap \im \pi$$. Then there is $$w\in V$$ such that $$v=\pi (w)$$ since $$v\in \im \pi$$. But $$v\in \ker \pi$$ also so that

\begin{equation*} 0=\pi (v)=\pi (\pi (w))=\pi (w)=v. \end{equation*}

Thus $$\ker \pi \cap \im \pi =\set {0}$$ so it remains to show that $$V=\ker \pi +\im \pi$$. For this, write $$v=(v-\pi (v))+\pi (v)$$. The second summand is certainly in $$\im \pi$$ while

\begin{equation*} \pi (v-\pi (v))=\pi (v)-\pi (\pi (v))=\pi (v)-\pi (v)=0 \end{equation*}

so the first is in $$\ker \pi$$ and we are done.

• 4.

• (a) All these sums are direct as each $$U_i\cap U_j=\set 0$$.

• (b) Note that $$(2,3,1)=(1,2,0)+(1,1,1)$$ and so can be written in two different ways as a sum $$u_1+u_2+u_3$$, with each $$u_{i}\in U_{i}$$:

\begin{gather*} (1,2,0)+(1,1,1)+(0,0,0)\\ (0,0,0)+(0,0,0)+(2,3,1). \end{gather*} Thus $$U_1+U_2+U_3$$ is not a direct sum.

This shows us that $$U_i\cap U_j=\set 0$$, $$i\neq j$$, is not enough to force $$U_1+U_2+U_3$$ to be direct.

• 5.

• (a) This is false: take $$V_1,V_2,V_3\leq \R ^2$$ to be the subspaces spanned at $$(1,0)$$, $$(0,1)$$ and $$(1,1)$$ respectively. Then any $$V_i+V_j=\R ^2$$ and $$V_i\cap V_j=\set {0}$$, for $$i\neq j$$. Now the left side is $$V_1+\set {0}=V_1$$ while the right is $$\R ^2\cap \R ^2=\R ^2$$.

• (b) This is also false. With the same $$V_i$$ as in part (a), the left side is $$V_1\cap \R ^2=V_1$$ while the right is $$\set {0}+\set {0}=\set {0}$$.

• (c) This is true: $$V_2,V_3\leq V_2+V_3$$ so that $$V_1\cap V_2, V_1\cap V_3\leq V_1\cap (V_2+V_3)$$. It now follows from Proposition 2.1 that $$(V_1\cap V_2)+(V_1\cap V_3)\subseteq V_1\cap (V_2+V_3)$$.

• 6.

• (a) No: a given $$V_2$$ has many complements. For example, take $$V=\R ^2$$, $$V_2$$ to be spanned by $$(1,0)$$ and then $$V_1,V_1'$$ to be spanned by $$(0,1)$$ and $$(1,1)$$ respectively.

• (b) This is true. For example, consider the projection $$\pi _1$$ with image $$V_1$$ and kernel $$V_2$$ and restrict this to $$V_1'$$ to get a linear map $$V_1'\to V_1$$. Then $$\ker (\pi _1{}_{|V_1'})=\ker \pi _1\cap V_1'=V_2\cap V_1'=\set {0}$$ so that $$\pi _1{}_{|V_1'}$$ injects. Moreover, for $$v_1\in V_1$$, write $$v_1=v_1'+v_2$$ with $$v_1'\in V_1'$$ and $$v_2\in V_2$$. Then $$v_1=\pi (v_1)=\pi _1(v_1'+v_2)=\pi _1(v_1')$$ so that $$\pi _1{}_{|V_1'}:V_1'\to V_1$$ surjects also and so is an isomorphism.