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anonymous
 4 years ago
Anyone good with changing basis!!!! I neeeeeed Help! :(
anonymous
 4 years ago
Anyone good with changing basis!!!! I neeeeeed Help! :(

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anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0no changing basis of a transformation matrix.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0the 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?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0my 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.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0(a) First column: 1, 0 Second column: 0, 0

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0The 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}.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Now, 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).

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0(where {e1,e2} is the canonical standard basis for R2)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0but how to i change the basis to be working in the standard basis instead of basis B?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0So you have to ask yourself: How do I write (1,0) in terms of linear combinations of v1 and v2?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Note that v1+2v2 = (3,0), so 1/3*v1+2/3*v2=e1

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0You do the same for e2.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Then you can calculate T(e1), T(e2) in terms of the standard basis and write down the matrix for (b).

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Btw. this is equivalent to finding the inverse matrix for the matrix with columns v1, v2.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i 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?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0a basis for B? What do you mean by that? B IS a basis.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0When 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.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0And then these coefficients are written down in a nicely ordered way, which is called a matrix. I.e. the ith column contains in the jth row the coefficient in front of b_j when you write T(b_i) in terms of the basis {b_1,...,b_n}.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i 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
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