Solving Linear Transformation Problem: L((7,5))^T Value Calculation

[electricalcoolness]

I have a question regarding a math problem that I do not know how to go about solving.

Let L: R^2 ---> R^2 be a linear operator. If L((1,2)^T)) = (-2,3)T
and L((-1,1)) = (5,2)^T determine the value of L((7,5))^T

Any insight would be much appreciated.

 

[calvino]

L((1,2)^T)) = (-2,3)T

Just to make things clear, L is your function, and T is what in this case? ...And..what class is this for?

 

[electricalcoolness]

T mean Transposed, sorry I made a typo.


Let L: R^2 ---> R^2 be a linear operator. If L((1,2)^T)) = (-2,3)^T
and L((-1,1)) = (5,2)^T determine the value of L((7,5))^T

This is some Linear Algebra homework I am stuck on.

 

[calvino]

T mean Transposed, sorry I made a typo.
Let L: R^2 ---> R^2 be a linear operator. If L((1,2)^T)) = (-2,3)^T
and L((-1,1)) = (5,2)^T determine the value of L((7,5))^T
This is some Linear Algebra homework I am stuck on.

k. Well, I'll assume (-1,1) is transposed also.

One way you can do this is by looking at the transformation matrix of L. Let's say its [x y]. Matrix multiplying ur vector by the transformation matrix should get you your answer. In this case, the values of x, y are not given but the answers are (by answers, i mean images). You should see that this becomes a problem of solving two equations.

 

[electricalcoolness]

I still a little confused, please can you make it a little more clear?

 

[shmoe]

Try writing (7,5) as a linear combination of (1,2) and (-1,1). How will this help?

PS. it's ok to think of your vectors as row vectors, you could then leave out the transpose. Makes things a little neater in text.

 

[calvino]

shmoe's idea is on the right track...sorry, but I was way off, i think.

 

[electricalcoolness]

Try writing (7,5) as a linear combination of (1,2) and (-1,1). How will this help?

I did, but I still don't see how it would help?

 

[calvino]

well now that you've done that consider the definition of A linear transformation.

A function L: R^n--->R^m is called a linear transformation or linear map if it satisfies

i) L(u+v)= L(u) + L(v) for all u,v in R^n
ii) L(cv)= cL(v) for all v in R^n, and scalar c

Using both, this defintion and the combination you just made you should be able to get your answer.

 

[electricalcoolness]

I just realized that the combination I created is one of a few different combinations, does that matter which combination I use? I still can't get an answer. Or rather, I still can't get the answer that matches the books. is my linear combination correct?
where x1 = (1,1) and x2 = (2,-1)

4*x1 + 3*x2 = 7
3*x1 + (x2) = 5

where x1 and x2 have been taken from the (1,2) and (-1,1)I might have figured something out,
if I allow for a matrix multiplied by some other matrix, is that how i come about my answer?

 

[electricalcoolness]

I really thank you guys for your help and patience with me.

I think I figured out my answer, and it all makes sense. You guys are awsome.