Okay so you know that
and
Now use the linearity of the operator to get
and
This gives a system of equations that you can solve for
After you solve for them you should get
These are the columns of the matrix representation of the transform.
Assume that R2>R2 is a linear transformation such that
T|1| =|1|
|2| |-3|
and
T|3| =|2|
|-1| |1|
Find the standard matrix of T.
Ok, so I get that |1,2|=e1+2e2 and that |3,-1|=3e1-e2, but what do I do from there on? It shows in the solutions that
e1=1/7|1,2| + 2/7|3, -1|
e2=3/7|1,2|-1/7|3,-1|
I understand that the are using the same thing from e1+2e2 and 3e1-e2, but where is the divisor of 7 coming from?
Sorry if it's formatted badly, I can scan the original page? Thanks
Thanks for the reply but I'm still confused. I don't get where you said it gives a system of equations for T(e1) and T(e2). Where would I get that from?
is it just
1 -1
2 1
Which I'm pretty sure is wrong... could you maybe put in the step of where you got the system of equations, and where from?
Thanks a lot
Let's take the following as given:
T is completely determined once the images of any basis for R^2 are known. *
With * in mind, I think I would read the initial statement, i.e.,
that T(1,2) = (1,-3) and T(3,-1) = (2,1), as telling me that
with respect to the basis B = {(1,2), (3,-1)} T is uniquely determined.
In this context the matrix is the matrix representation of T
relative to basis B. Call it M'.
But, the problem asks you to find the "standard matrix" of T.
I read that as meaning the matrix representation of T relative to
the standard (or natural) basis E = {(1,0),(0,1)}.
So I think there's more work to do.
You already have M'.
Can you say what the matrix might represent?
Maybe the matrix of transition from E to B?
What about ?
Hi, I don't know if its because it's late, or because I'm just really stupid, but I'm still really lost.
To solve, do you mean in this fashion:
>>
so
Just a wild guess here
I completely get this:
and
and I also get that you have to use
and
where the e1 and e2 are (1,2) and (3,-1) respectively, but how do you find out what T is for each?
From what I'm looking at, the multiple stays the same, but there is a divisor for each T, in this case 7.
How do I solve for that T? Hopefully I'm being clear enough
EDIT I GOT IT, I guess I just needed some sleep Thanks guys, makes sense now