Hard new stuff

Question 1

Given ANY linear map A V->W, how do you make a matrix for it?

Answer 1

1. Determine bases for both the domain and the codomain.
   ex. B = {v1, …, vn} is a basis for the domain
   C = {w1, …. , wm} is a basis for the codomain.
2. For each element in the basis of the domain, determine the vector representation of that element with respect to the codomain ([Avi]c).
   Place each of these in the columns in the matrix, and you done!

Question 2

What is a linear map?

Answer 2

A function from V->W (both are vector spaces) for which:
1. f(v1 + v2) = f(v1) + f(v2)
2. alphaf(v) = f(alphav)

Question 3

Given a linear map from V -> W, what will the dimensions of the matrix be?

Answer 3

dim W * dim V

Question 4

What does this mean: [Avi]c

Answer 4

Given A, a linear map V-> W, where
B = {v1,… vn} is a basis for V
and C = {w1, …. wm} is a basis for W,

[Avi]c is the representation of some the output of the map when you plug in vi, some element of the basis, in terms of the elements of the basis of W.

So for instance, if A(v1) = w2 - 2w3…
[Avi]c = (a vector) (0 1 -2 0 … 0)

Question 5

What does this mean: [A]c,B

Answer 5

the matrix representation of linear map A with respect to codomain basis C and domain basis B

Question 6

What does this mean? Why is it true?
[Av]c = [A]c,B *[v]B

Answer 6

[v]B is a representation of the input vector in terms of the elements of the domain basis.
[A]c,B : each col of this is a representation of the OUTPUT ASSOCIATED WITH each input basis vector in terms of the basis of the output space.

multiplying by [v]B says “Ill take one of this output associated with my first basis vector, one of this output associated with second basis vector, 2 of etc.”

Question 7

How to determine the grid for laplace expansion

Answer 7

+ in top left, - everywhere else

Question 8

Laplace expansion

Answer 8

1. Pick your favorite row or col.
2. The determinant = (sum for all entries){ signentry(determinantOfA with row, col of entry eliminated)}

Question 9

Rank

Answer 9

Dimension of the image (output space)

Question 10

Nullity

Answer 10

Dimension of the kernel (inputs that send to zero space)

Question 11

Rank and Nullity theorem. What is it, and why is it true?

Answer 11

Theorem: the rank and nullity of a matrix sum to the number of columns.

Reason: the nullity is the dimension of the null space. If this is not zero, it will be equal to the number of free variables.

The rank is the dimension of the image space. this is going to be the number of non-free variables

Question 12

The number of columns of the matrix representation of a linear map corresponds to …

Answer 12

The dimension of the codomain

Question 13

The number of rows in The matrix representation of a linear map corresponds to

Answer 13

The dimension of the domain