Finding the Image of a Vector under a Linear Transformation

[superdave]

Homework Statement

Let L: R^3 -> R^3 be a linear transformation such that

L(i) = [1 2 -1], L(j) = [1 0 2] and L(k) = [1 1 3].

Find L([ 2 -1 3)].

All the numbers in [ ] should be vertical, but I don't know how to set that up.

Homework Equations

The Attempt at a Solution

I'm not sure how to even approach this. I've tried looking at examples in the text and they aren't clear.

I would think that you make a matrix L with the three columns i j k as above. and x = [2 -1 3] and just calculate Lx=b to find b.

But a similar problem in the text has a single number answer. I would've guessed that it gives a vector.

 

[radou]

You know how the transformation acts on the basis {i, j, k}. Let v be any vector, then v = αi + βj + γk. Then L(v) = αL(i) + βL(j) + γL(k).

The vectors L(i), L(j) and L(k) are the columns of the matrix representation of the operator L.

 

[Office_Shredder]

But a similar problem in the text has a single number answer. I would've guessed that it gives a vector.

You should get a vector here. Maybe you should post the similar problem and we can help you figure out what the difference is?