Solve the Math Equation: 1. (b)? 2. (c) -2 5. (a)

[Ted123]

Homework Statement

[PLAIN]http://img411.imageshack.us/img411/6141/48919675.jpg
[PLAIN]http://img444.imageshack.us/img444/5839/55504929.jpg
[PLAIN]http://img574.imageshack.us/img574/2935/26916604.jpg
[PLAIN]http://img560.imageshack.us/img560/189/87892973.jpg
[PLAIN]http://img148.imageshack.us/img148/5259/11201645.jpg

The Attempt at a Solution

1. (b)
2. ?
3. (c)
4. -2
5. (a)

 

[Mark44]

You'll probably get more responses if
a) you post one problem at a time, and
b) you give some explanation of why you picked a given choice.

 

[Ted123]

You'll probably get more responses if
a) you post one problem at a time, and
b) you give some explanation of why you picked a given choice.

Well really it's just 2 and 5 that I'm unsure about.

My answer to 1 should be (a) and 3 and 4 are correct.

For 2, the rank of M = rank of $\phi$ but what is the rank of $\text{Ker}(\phi)$ ?

 

[micromass]

Do you know any dimension-formula connecting the kernel and the image?

 

[Ted123]

Do you know any dimension-formula connecting the kernel and the image?

Rank-nullity formula: dim(V) = dim[ker($\phi$)] + dim[Im($\phi$)]

 

[micromass]

Yes, use that to solve your problem.

 

[Ted123]

Yes, use that to solve your problem.

Of course, dim[Im($\phi$)] = rank($\phi$)

So 15 = dim[ker($\phi$)] + 5 so dim[ker($\phi$)] = 10 ?

How about 5?

 

[micromass]

Well, 5(a) is certainly correct, you're right about that.
But there are more statements in 5 that are correct!

 

[Ted123]

Well, 5(a) is certainly correct, you're right about that.
But there are more statements in 5 that are correct!

5(c) true and 5(b) false ?

 

[micromass]

Can you give me a counterexample for 5(b)?
And can you motivate why 5(c) is true for you?

 

[Ted123]

Can you give me a counterexample for 5(b)?
And can you motivate why 5(c) is true for you?

M is invertible so 5(b) is true and 5(c) is false

 

[micromass]

Can you provide any motivations for this?

 

[Ted123]

Can you provide any motivations for this?

Well I know that 5(b) is right as M has to be invertible.

For 5(c),

rank(M) = rank($\phi$) = dim[Im($\phi$)]

dim(V) = dim(W) since isomorphic finite-dimensional vector spaces have the same dimension.

So applying the rank-nullity formula, we see that

dim(V) = dim(W)= dim[ker($\phi$)] + rank($\phi$)

But does dim[ker($\phi$)] = 0 ?

 

[micromass]

Well, $$\phi$$ is invertible. Does that imply that the kernel is trivial?

 

[Ted123]

Well, $$\phi$$ is invertible. Does that imply that the kernel is trivial?

Yes

so 5(c) is true.

 

[micromass]

Correct. So all statements in 5 are correct!