How Do You Transform the Matrix of a Linear Map Between Different Bases?

[JamesGoh]

Homework Statement

Suppose that T : R2 → R2 is linear and has matrix

-2 1
5 2

with respect to the standard basis S of R2.

B = 1 1
5 6

(B is another poorly constructed matrix)

What is the matrix of T with respect to B?

Homework Equations

T$_{C,B}$ = (T$_{C,S}$)$^{-1}$I$_{B,S}$

The Attempt at a Solution

Please see the pdf called q6b

Also to see the question better presented look at problems1.pdf. Go to Problem sheet 3 question 6b

 

[JamesGoh]

sorry guys, I should point out that

the answer to the question should be

3 7
0 -3

which I don't seem to be getting

 

[ehild]

T$_{C,B}$ = (T$_{C,S}$)$^{-1}$I$_{B,S}$

That is wrong. A matrix in a new basis B is I_BS^-1T I_BS

ehild

 

[JamesGoh]

When you say T, do you mean T with respect to the standard basis ?

 

[JamesGoh]

That is wrong. A matrix in a new basis B is I_BS^-1T I_BS

ehild

In terms of basis reference, does your formula do the following (in terms of matrix composition)

1. You create the identity map of matrix B with respect to the standard basis

2. You multiply the B identity map with T to create a new matrix with respect
to the standard basis ?

3. You multiply the result of step 2 with the inverse of the identity map in step 1 to get the answer with respect to B

 

[ehild]

With respect to the standard basis T=T_s, and T_b=I_BS-¹T_s I_BS is the matrix of the linear transformation in the new basis.

Think: The new basic vectors b₁, b₂ are B times the standard basic vectors. Edit: With "B"I denoted the matrix with columns equal to the basic vectors b₁ and b₂. And you get beck the standard basic vectors by multiplying b₁, b₂ by B^-1.
T transforms the basic vectors of B into vectors Tb₁ and Tb₂, defined in the standard basis. You need to express these vectors in terms of the new basis, so apply B^-1 to them.

 

[ehild]

In terms of basis reference, does your formula do the following (in terms of matrix composition)

1. You create the identity map of matrix B with respect to the standard basis

2. You multiply the B identity map with T to create a new matrix with respect
to the standard basis ?

3. You multiply the result of step 2 with the inverse of the identity map in step 1 to get the answer with respect to B

You explained it very well. Yes. that is what I wanted to do, but I could not express myself so well.

ehild

 

[HallsofIvy]

Just a word about notation: It does not really make sense to ask about the matrix of a linear transformation "with respect to" another matrix. In your attached pdf files, B is NOT a matrix, it is two vectors, a new basis for $R^2$.

Another way to find the matrix of a linear transformation with respect to a given (ordered) basis is: Apply the linear transformation to the each basis vector in turn. Write the result as a linear combination of the basis vectors. The coefficients give the columns of the matrix.

 

[ehild]

Thanks, HallsofIvy. I know that I cannot mix vectors with transformations...I just used that the matrix that transforms the standard basis into a new basis has columns equal to the new basic vectors.
I hope, I am right... Being a physicist, I use Maths a bit sloppy way.

 

[Ray Vickson]

Homework Statement

Suppose that T : R2 → R2 is linear and has matrix

-2 1
5 2

with respect to the standard basis S of R2.

B = 1 1
5 6

(B is another poorly constructed matrix)

What is the matrix of T with respect to B?

Homework Equations

T$_{C,B}$ = (T$_{C,S}$)$^{-1}$I$_{B,S}$

The Attempt at a Solution

Please see the pdf called q6b

Also to see the question better presented look at problems1.pdf. Go to Problem sheet 3 question 6b

If the standard basis is e1=[1 0]^t and e2=[0 1]^t (^t = transpose) and b1=[1 5]^t, b2 = [1 6]^t, you can express e1 and e2 as linear combinations of b1 and b2, just by solving the equations b1=e1+5*e2 and e2=e1+6*e2 for e1 and e2; you can do this just as though the e's and b's were real variables instead of vectors---the algebra does not care what they are. A vector v = x1*b1 + x2*b2 can thus be written in terms of basis {e1,e2}, say as y1*e1+y2*e2, with y1,y2 = known linear combinations of x1,x2. Applying the matrix A = [[-2 1],[5 2]]to v = y1*e1+y2*e2 gives (-2y1+y2)*e1+(5y1+2y2)*e2. Now back-substitute for the yi in terms of the xi and for the ej in terms of the bj. You should get the result Tv = (3x1+7x2)*b1+(-3x2)*b2. Therefore, the matrix representation of T in the basis {b1,b2} is [[3 7],[0 -3]]. Once you have grasped these concepts through some simple examples, done step-by-laborious-step, then you can more successfully see where the matrix formulae come from and what they mean.

RGV