Linear Algebra (Matrix representation of linear operators)

[Treeline]

Homework Statement

Determine [T]_β for linear operator T and basis β

T:((x₁; x₂]) = [2x₁ + x₂; x₁ - x₂]

β = {[2; 1], [1; 0]}

Homework Equations

Now that would be MY question

The Attempt at a Solution

Well the answer is [1, 1; 3, 0], but i have no idea what I'm even supposed to be doing here.
I have my book to refer to, but it's like something has been left out, assuming I'd understand.
I don't. Can anyone take a qualified guess as to what I do here? I'm at a loss.

Thanks

 

[Mark44]

Homework Statement

Determine [T]_β for linear operator T and basis β

T:((x₁; x₂]) = [2x₁ + x₂; x₁ - x₂]

β = {[2; 1], [1; 0]}

Homework Equations

Now that would be MY question

The Attempt at a Solution

Well the answer is [1, 1; 3, 0], but i have no idea what I'm even supposed to be doing here.
I have my book to refer to, but it's like something has been left out, assuming I'd understand.
I don't. Can anyone take a qualified guess as to what I do here? I'm at a loss.

Thanks

Start by looking in your book for the meaning of this notation: [T]_β. Understanding what it means should go a long ways toward being able to do something in this problem.

 

[Deveno]

a matrix A is a rectangular array of field elements (m rows, n columns).

a linear operator (transformation) is a linear function from one vector space to another.

but...since we can write vectors as an array (1 row, or 1 column, take your pick), once we choose a basis (so the entries in the array are the coordinates of our vector in our basis), it seems that since matrices turn (via matrix multiplication, if we multiply the vectors on the right side) nx1 arrays into mx1 arrays (that is, n-vectors into m-vectors), perhaps we can turn a linear operator into a matrix.

that is, we want to turn a linear function into a collection of numbers. HOW we do so, depends on our choice of bases for the vector spaces involved (different bases give different matrices).

what you haven't told us, is what spaces T operates on, and what field you are using. there's actually lots of possible fields, although most beginning courses stick to the real numbers, because they are very familiar. my guess is that:

T:R²→R², which is a very important piece of information that you have not told us.

what size matrix takes a 2-vector to a 2-vector? well, it must be 2x2. let's do some detective work.

since [T]_β is a 2x2 real matrix, [T]_β =

$$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$

what we need to do is figure out a,b,c and d. but we need to use the basis β somehow. and this is where the problem is sort of sneaky. for example (x₁,x₂) is already "coordinatized" by a basis already, but this basis is "invisible", because it's the standard basis for R². the upshot of this, is that x₁ and x₂ are "the wrong numbers" (coordinates), because we need to work with the basis β.

we need to find the β-coordinates of (x₁,x₂).

let's do a somewhat easier problem first. we know that if e₁ = (1,0), and e₂ = (0,1), then (x₁,x₂) = x₁e₁ + x₂e₂. this is how linear combinations are turned into coordinate arrays. so what we want to do is write:

e₁ = (1,0) = a₁(2,1) + a₂(1,0).

since β is a basis for R², there's only ONE way to do this. and it's fairly obvious that a₁ = 0, a₂ = 1. that is:

e₁ = [0,1]_β. next we want to write:

e₂ = (0,1) = b₁(2,1) + b₂(1,0) = (2b₁+b₂,b₁).

again, it's fairly obvious that b₁ = 1, so b₂ must be -2. so now we know that e₂ = [1,-2]_β.

now we are in a position to write (x₁,x₂) in β-coordinates:

(x₁,x₂) = x₁e₁ + x₂e₂ = x₁[0,1]_β + x₂[1,-2]_β = [x₂,x₁-2x₂]_β.

now [1,0]_β = 1(2,1) + 0(0,1) = (2,1). and by definition (or just by doing the matrix multiplication, writing [1,0]_β as a column vector), the first column of [T]_β is

[T]_β([1,0]_β)^T.

but if [T]_β is really supposed to represent the linear transformation T, then this ought to be [T(2,1)]_β. can you continue?

 

[Treeline]

After looking at this piece of (*) for 3 hours over 2 days, i finally solved it, and i can reproduce the calculations - And actually somewhat understand the logic.

Deveno, you really helped a lot, and your post was very informative. Thanks a LOT for the effort, and Merry Christmas to the both of you! :)

Note: T:R2→R2 was indeed the space, although even the assignment forgot to mention. I guess, as it is indeed the beginning course, this was just left out and to be assumed.

Again, thanks a lot!

 

[Mark44]

Note: T:R2→R2 was indeed the space, although even the assignment forgot to mention. I guess, as it is indeed the beginning course, this was just left out and to be assumed.

It wasn't explicitly stated that the transformation T maps R² to R², but the definition of T makes this very clear.

T:((x₁; x₂]) = [2x₁ + x₂; _x1 - x₂]

The inputs are vectors in R², as are the outputs.

 

[Treeline]

Thanks for clarifying! :)