Question

Anyone good with changing basis!!!! I neeeeeed Help! :(

Answer 1

logs?

Answer 2

no changing basis of a transformation matrix.

Answer 3

the transformation matrix is made up of the basis vectors as columns, transformation is applied by multiplying with the transformation matrix, where is the part that's troubling you?

Answer 4

my problem is here:
Suppose that vectors v1 = (1, 2), v2 = (2,−1), and that the basis B is B=v1, v2 . (this is a list)
Let T be the linear transformation from R2 to R2 given by T(v1) = v1 and T(v2) = 0.
(a) Write down the matrix for T in the new basis B. (You should be able to do this
directly from the definition of T).
(b) Use this to write down the matrix for T in the standard basis.

Answer 5

(a) First column: 1, 0
Second column: 0, 0

Answer 6

The first column has to contain the coefficients of T(v1) in the new basis {v1,v2}.
The second column contains the coefficients of T(v2) in the new basis {v1,v2}.

Answer 7

Now, for (b) you have to express T(e1) in terms of the basis {e1,e2} and that will be your first column, and the second accordingly for T(e2).

Answer 8

(where {e1,e2} is the canonical standard basis for R2)

Answer 9

but how to i change the basis to be working in the standard basis instead of basis B?

Answer 10

So you have to ask yourself: How do I write (1,0) in terms of linear combinations of v1 and v2?

Answer 11

Note that v1+2v2 = (3,0), so 1/3*v1+2/3*v2=e1

Answer 12

You do the same for e2.

Answer 13

Then you can calculate T(e1), T(e2) in terms of the standard basis and write down the matrix for (b).

Answer 14

Btw. this is equivalent to finding the inverse matrix for the matrix with columns v1, v2.

Answer 15

Do you understand?

Answer 16

i understand for the most part, until applying the transformation, becuase if i'm applying that transformation aren't i still working in a basis for B?

Answer 17

a basis for B? What do you mean by that? B IS a basis.

Answer 18

When you are asked to find a matrix of a linear transformation T of Rn to Rn with respect to a certain basis {b_1,b_2,...,b_n}, then this just means you are supposed to find the coefficients of the images T(b_i) in terms of the given basis. So you are supposed to express the images of the basis vectors in terms of the basis.

Answer 19

And then these coefficients are written down in a nicely ordered way, which is called a matrix. I.e. the i-th column contains in the j-th row the coefficient in front of b_j when you write T(b_i) in terms of the basis {b_1,...,b_n}.

Answer 20

i know but thats what i mean, if i apply the transformation matrix to the standard basis vectors arent we still working in that basis? when for part b what we're supposed to do is put the transformation matrix into standard basis for instead of B basis