Few question on exam review

[warwick]

I'm not sure how to tackle #6.

I want to make sure my logic about the rest is correct. There's still a little fuzziness on how to show something is a subspace. I know it has to have the zero vector and be closed under scalar multiplication and addition.

 

[warwick]

Can anyone help? I'm not sure how to do number 6 and 18, and I want to make sure I did the others correctly.

 

[daon]

For (6), I'll do (a):

(u+v)+w = (<u1, u2> + <v1, v2>) + <w1, w2> = <u1+v1, u2+v2> + <w1, w2> = <(u1+v1)+w1, (u2+v2)+w2>

(...the following equality comes from the fact that we are now working with real numbers, which are, of course, associative.)

= <u1 + (v1 + v2), u2 + (v2+w2)> = <u1, u2> + <v1 + w1, v2+w2> = <u1,u2> + (<v1, v2>+<w1+w2>) = u + (v+w).



For 18, try to show that if A=A^t and B=B^t then (A+B)^t = A+B. Then if c is real, show (cA)^t = cA.

 

[warwick]

For (6), I'll do (a):

(u+v)+w = (<u1, u2> + <v1, v2>) + <w1, w2> = <u1+v1, u2+v2> + <w1, w2> = <(u1+v1)+w1, (u2+v2)+w2>

(...the following equality comes from the fact that we are now working with real numbers, which are, of course, associative.)

= <u1 + (v1 + v2), u2 + (v2+w2)> = <u1, u2> + <v1 + w1, v2+w2> = <u1,u2> + (<v1, v2>+<w1+w2>) = u + (v+w).



For 18, try to show that if A=A^t and B=B^t then (A+B)^t = A+B. Then if c is real, show (cA)^t = cA.

But, how vector addition is defined is throwing me off. (x1, x2) + (y1, y2) = (x1 + y2, x2 +y1)

Also, in the last line, shouldn't it be = <u1 + (v1 + w1), u2 + (v2+w2)> ?

 

[daon]

Whoops. I didn't see that the definition was so different (and yes to your question, I copied incorrectly).

So You'll do:

u+(v+w) = <u1, u2> + (<v1, v2> + <w1, w2>) = <u1, u2> + <v1 + w2, v2 + w1> = <u1 + (v2 + w1), u2 + (v1 + w2)> = <(u1 + v2 + w1), (u2 + v1 + w2)>

(u+v)+w = (<u1, u2> + <v1, v2>) + <w1, w2> = <u1 + v2, u2 + v1> + <w1, w2> = <(u1 + v2) + w2, (u2 + v1) + w1> = <(u1 + v2 + w2), (u2 + v1 + w1)>

These are not the same.

 

[warwick]

Whoops. I didn't see that the definition was so different (and yes to your question, I copied incorrectly).

So You'll do:

u+(v+w) = <u1, u2> + (<v1, v2> + <w1, w2>) = <u1, u2> + <v1 + w2, v2 + w1> = <u1 + (v2 + w1), u2 + (v1 + w2)> = <(u1 + v2 + w1), (u2 + v1 + w2)>

(u+v)+w = (<u1, u2> + <v1, v2>) + <w1, w2> = <u1 + v2, u2 + v1> + <w1, w2> = <(u1 + v2) + w2, (u2 + v1) + w1> = <(u1 + v2 + w2), (u2 + v1 + w1)>

These are not the same.

Ok. Here's my new work. Tell me if I'm right. The key says I am, but I just want to make sure.

 

[daon]

Your answers for b and c look correct, but if your teacher/professor is a stickler about the axiomatic approach, more work and explanation should be shown.

 

[warwick]

Your answers for b and c look correct, but if your teacher/professor is a stickler about the axiomatic approach, more work and explanation should be shown.

What else would I say?

For 18, the key says the dimension is three. I see I can find two matrices - [(1,0), (0,1)], [(0,1), (1,0)].

 

[daon]

Look at how I did (a), I took steps some would consider "obvious" and not do. When working with obscure definitions, its best to do it that way. Of course this is just my opinion.

It would appear that the dimension was 2 as you have written it. However what about a matrix like this:

[a, 0]
[0, 0]

 

[warwick]

Look at how I did (a), I took steps some would consider "obvious" and not do. When working with obscure definitions, its best to do it that way. Of course this is just my opinion.

It would appear that the dimension was 2 as you have written it. However what about a matrix like this:

[a, 0]
[0, 0]

Yeah, I didn't fully write everything out like you did. I will on the exam, though.

18 just asks about symmetric matrices. With the suggestions you gave me earlier for this problem, I'll use that [(a,b), (b,a)] to show that the matrices are closed under vector addition and scalar multiplication and also has the zero vector. That will show that W is a subspace of M22. I don't know about the matrix you just put. I suppose a 2x2 zero matrix is indeed a symmetric matrix.

 

[daon]

Your assumption on the symmetric matrix is a little off.

A 2x2 Symmetric matrix is of the form:

[a b]
[b c]

A 3x3 symmetric matrix is of the form:

[a b c]
[b d e]
[c e f]

The entry \(\displaystyle a_{ij}\) must equal the entry \(\displaystyle a_{ji}\) for \(\displaystyle A=A^t\).

...

Had the subset been defined as you assumed, your basis would be correct.