Linear transform questions

[Kaspelek]

Also just working on another question, especially stuck with the last part.

It's basically definitions.



This is what i've got so far, correct me if I'm wrong.

a) k components, k components.
b) R^n to R^n
c) R^rank(T)
d)R^nullity(T)
e) Completely unsure (need help with this)

 

[TheBigBadBen]

Also just working on another question, especially stuck with the last part.

It's basically definitions.

View attachment 825

This is what i've got so far, correct me if I'm wrong.

A few things wrong over here, let's get started:

a) k components, k components.

Remember how matrix multiplication works: the number of columns of the matrix on the left has to match the number of rows of the matrix on the right. In this case, A has d rows, which means that x (a column vector) has to have d components. The product, b, will have the same number of columns as x (one column) and the same number of rows as A (k rows). So, b has k components.

b) R^n to R^n

I don't see an "n" anywhere in this question. Since A takes a vector of d components and gives a vector of k components, A is a map from $\mathbb{R}^d$ to $\mathbb{R}^k$.

c) R^rank(T)
d)R^nullity(T)
e) Completely unsure (need help with this)

The answers to c) and d) are $\mathbb{R}^k$ and $\mathbb{R}^d$ respectively.

I'll put e) as it's own post.

 

[TheBigBadBen]

Now for e): Let's make a list for reference. For our purposes, I don't care about the difference between A and $T_A$ (which gives us a few answers for free, incidentally).

1. row space of A
2. kernel of A
3. column space of A
4. solution space of A
5. image of A
6. null space of A
7. rank of A
8. rank of A
9. dimension of image of A
10. nullity of A
11. nullity of A
12. dimension of the row space of A
13. dimension of the kernel of A
14. dimension of the image of A^T
15. nullity of A^T
16. number of rows of A
17. rank of A^T

The question is to "group all synonyms." Now I'm fairly sure the answer should be:
1
2,6
3,5
4
7,8,9,12,14,17
10,11,13
15
16

any questions on a particular grouping?