Linear Transform Synonyms: Grouping Components and Spaces

[Kaspelek]

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) 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?