Help with Answer Checking: Can you check my answers and help me?

[Ted123]

Can anyone check my answers/help me?

1
[PLAIN]http://img689.imageshack.us/img689/3912/59048711.jpg

My answer: 3

2
[PLAIN]http://img171.imageshack.us/img171/812/10085212.jpg

My answer: b,c

3
[PLAIN]http://img252.imageshack.us/img252/7706/81927762.jpg

My answer: False

4
[PLAIN]http://img573.imageshack.us/img573/5450/70607877.jpg

My answer: a

5
[PLAIN]http://img51.imageshack.us/img51/6527/12780304.jpg

My answer: ?

6
[PLAIN]http://img715.imageshack.us/img715/8680/30828400.jpg

My answer: ?

7
[PLAIN]http://img251.imageshack.us/img251/1006/48454960.jpg

My answer: a,e and ...?

 

[jbunniii]

1,2,3 are correct.

Your answer for 4 is partially correct: there are two statements that are definitely true, not one.

For 5, use this theorem:

dim(V) = dim(kernel(phi)) + dim(image(phi))

For 6, I am not sure what this notation means:

$$\mathbb{R}^1 0$$

Can you please clarify?

For 7, a) and e) are correct, but there is one more. What is the dimension of the quotient space V/U?

 

[Ted123]

1,2,3 are correct.

Your answer for 4 is partially correct: there are two statements that are definitely true, not one.

For 5, use this theorem:

dim(V) = dim(kernel(phi)) + dim(image(phi))

For 6, I am not sure what this notation means:

$$\mathbb{R}^1 0$$

Can you please clarify?

For 7, a) and e) are correct, but there is one more. What is the dimension of the quotient space V/U?

For 4 is c) correct too?

And for 5 how do I find the dimension of image($\phi$) ?

In question 6, I don't know what that notation means either, so I've sent an email and queried it.

For 7, d) is correct too (12 - 3 = 9)

 

[jbunniii]

For 4 is c) correct too?

Yes. Can you explain why?

And for 5 how do I find the dimension of image($\phi$) ?

This is given, implicitly. The map is a surjection (onto). So image($\phi$) = ?

For 7, d) is correct too (12 - 3 = 9)

Right.

 

[Ted123]

Yes. Can you explain why?

For $\phi$ to be an isomorphism it must be one-to-one which implies the kernel is trivial.

This is given, implicitly. The map is a surjection (onto). So image($\phi$) = ?

$\text{image}(\phi ) = W$ and $\text{dim}(W)=4$

so $\text{dim}[\text{kernel}(\phi )] = 8-4=4$

I've had an answer back for question 6 and apparently this should read $\phi : \mathbb{R}^{10} \to \mathcal{C}[0,1]$ (ie. $\mathbb{R}^{10}$ is the space of real column vectors of length 10).

 

[jbunniii]

Yes, those answers are correct.

OK, so for 6, you can again use

dim(V) = dim(kernel(phi)) + dim(image(phi))

Hint: the target space is irrelevant.

 

[Ted123]

Yes, those answers are correct.

OK, so for 6, you can again use

dim(V) = dim(kernel(phi)) + dim(image(phi))

Hint: the target space is irrelevant.

$\text{dim}[\text{image}(\phi)] = 10-1 = 9$

 

[jbunniii]

$\text{dim}[\text{image}(\phi)] = 10-1 = 9$

Correct.