Question

Let V be the set of 2x2 matrices with standard definitions for vector addition and scalar multiplication. Determine whether V is a vector space. If V is not a vector space, show that at least one of the 10 axioms does not hold.

Have to show that the set of all real symmetric matrices, that is, the set of all matrices such that A^t=A is a vector space.

Answer 1

So which axioms are you having trouble showing?

Answer 2

All of them. I just do not understand for some reason.

Answer 3

I can walk you through some of them, and hopefully you'll be able to finish the rest.

Answer 4

ok

Answer 5

Oh, before I start, we're assuming this space of symmetric matrices has real valued entries, and has 2x2 matrices. Correct?

Answer 6

yes

Answer 7

So if \(v\in V\), \(v\) looks like \[\begin{bmatrix}a&b\\b&a\end{bmatrix}\]for \(a,b\in\mathbb{R}\).

Axiom 1: If \(u,v\in V\), then \(u+v\in V\).
Pf: Let \[u=\begin{bmatrix}a&b\\b&a\end{bmatrix},\quad v=\begin{bmatrix}c&d\\d&c\end{bmatrix}.\]Then \[u+v=\begin{bmatrix}a+c&b+d\\b+d&a+c\end{bmatrix}.\]This is still a symmetric matrix, so axiom 1 is satisfied.

Make sense?

Answer 8

yes.

Answer 9

So then Axiom 2 would be satisfied right?

Answer 10

Almost. There's one more small thing we have to show.

Axiom 2: If \(u,v\in V\), then \(u+v=v+u\).
Pf: We know\[u+v=\begin{bmatrix}a+c&b+d\\b+d&a+c\end{bmatrix}=\begin{bmatrix}c+a&d+b\\d+b&c+a\end{bmatrix}=v+u.\]So axiom 2 is satisfied because addition of real numbers is commutative.

Answer 11

Axiom 3: If \(u,v,w\in V\), then \((u+v)+w=u+(v+w)\).
Pf: I think you can do this yourself. Just let \(u,v\) be the same matrices as above, and let \[w=\begin{bmatrix}e&f\\f&e\end{bmatrix}\]Then add the matrices together, and use the fact that addition of real numbers is commutative.

Answer 12

Axiom 4: There exists \(\vec{0}\in V\) such that \(\vec{0}+v=v+\vec{0}=v\) for all \(v\in V\).
Pf: Again, I think you can do this. Let \[\vec{0}=\begin{bmatrix}0&0\\0&0\end{bmatrix}\]and do the necessary operations.

Answer 13

yes those last two are starting to make sense now.

Answer 14

Sorry, for Axiom 3, I should have said "use the fact that addition of real numbers is associative."

Answer 15

I knew what you meant.

Answer 16

Axiom 5: For all \(v\in V\) there exists some \(-v\in V\) such that \(v+(-v)=-v+v=\vec{0}\).
Pf: Another very straightforward one. \[v=\begin{bmatrix}a&b\\b&a\end{bmatrix},\quad -v=\begin{bmatrix}-a&-b\\-b&-a\end{bmatrix}.\]Add them together.

Answer 17

Axiom 6: For all \(c\in\mathbb{R}\) and \(v\in V\), \(cv\in V\).
Pf: Let\[v=\begin{bmatrix}a&b\\b&a\end{bmatrix}.\]Then\[cv=c\begin{bmatrix}a&b\\b&a\end{bmatrix}=\begin{bmatrix}ca&cb\\cb&ca\end{bmatrix}.\]This is still in the form we want, so this axiom is satisfied.

Answer 18

ok starting to make sense now thanks.

Answer 19

I think you'll be able to handle the rest. Just make sure to take things one step at a time. Also, for the last axiom, use \[\vec{1}=\text{Id}_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}\]

Answer 20

Call me back if you need help with any more of the axioms.

Answer 21

I have another question for you.

Let
V=\[\left\{ \left(\begin{matrix}\cos t \\ \sin t\end{matrix}\right) \right\}\]
and define

\[\left(\begin{matrix}\cos t_1 \\ \sin t_1\end{matrix}\right)⊕ \left(\begin{matrix}\cos t_2 \\ \sin t_2\end{matrix}\right)=\left(\begin{matrix}\cos (t_1+t_2) \\ \sin (t_1+t_2)\end{matrix}\right)\]

c ⨀\[\left(\begin{matrix}\cos t \\ \sin t\end{matrix}\right)=\left(\begin{matrix}\cos ct \\ \sin ct\end{matrix}\right)\]


Show that if ⊕ and ⨀ are the standard componentwise operations, then V is not a vector space.

I'm not quite sure it is not a vector space.

Answer 22

I think your problem will be with axiom 5: For all \(v\in V\), there exists a \(-v\) such that \(-v+v=v+(-v)=\vec{0}\).

There is a \(\vec0\) in this space (try to find this), but you may not have the inverses.