Change of basis?

Question

So the question is...

A transformation $T$ is denoted by $T(x,y)=(x+y,x-y)$.

$C$ is the basis $\{(1,-1),(1,1)\}$

$D$ is the basis $\{(1,2),(1,0)\}$

I know $T(C)=\{(0,2),(2,0)\}$

But how do I express that in terms of $D$ and more importantly what would the matrix representation of $ [T]_C^D $ look like?

Some formatting help would also be appreciated. Thanks!

Yeah but what would [T]cD be
Idk how to format (new to this) but C would be the subscript and D would be the superscript next to [T] — user139985, Apr 18 '14 at 17:30
$ [T]_C^D $ would give: $ [T]C^D $. If you wanted $ [T]{C^D} $ then, $ [T]_{C^D} $ — hjpotter92, Apr 18 '14 at 17:31
Do you know how to do change of basis for transformations like $T(u,v) = (u,v)$, in other words, the usual kind of vector? Just do that, and let $u = x+y, v = x-y$. — Emily, Apr 18 '14 at 17:40

score 2 · Answer 1 · answered Apr 18 '14 at 17:38

I won't do the computation but I will give you something much better -- the recipe of how to find what you are looking for:

Note that in the standard basis $(1,0), (0,1)$ the matrix of $T$ is $$ M = \begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}$$

This is a map taking a vector represented in the standard basis to a vector represented in the standard basis.

What you want is a matrix that takes a vector in the basis $C$ into a vector in the basis $D$. So given a vector in the basis $C$ you could transform it into a vector in the standard basis, apply $T$ and transform it into the basis $D$.

If $M_C$ denotes the matrix transforming from $C$ into the standard basis and $M_D$ from the standard basis into $D$ then the matrix you want will be $M_DMM_C$.

Now all that's left is to compute $M_C$ and $M_D$. But that's easy: From $C$ into the standard basis will be the matrix with columns $(1,-1), (1,1)$ (that's the basis vectors of $C$). For the other matrix take the inverse transform from $D$ into the standard basis and compute its inverse.

Change of basis?

1 Answers1