Linear Algebra (Matrix representation of linear operators)

In summary: Can someone point me in the right direction?In summary, the student is trying to solve a homework equation, but is having trouble understanding what he is supposed to be doing. He has a book to help him, but something is not working right. He needs to find the β-coordinates for (x1,x2), which is a task that is fairly easy with the help of the basis β. After some algebra, he is able to solve for (x1,x2) in β-coordinates.
  • #1
Treeline
3
0

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 :rolleyes:


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 :smile:
 
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  • #2
Treeline said:

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 :rolleyes:


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 :smile:

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.
 
  • #3
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 =

[tex]\begin{bmatrix}a&b\\c&d\end{bmatrix}[/tex]

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?
 
Last edited:
  • #4
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! :smile:
 
  • #5
Treeline said:
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.

Treeline said:
T:((x1; x2]) = [2x1 + x2; x1 - x2]
The inputs are vectors in R2, as are the outputs.
 
  • #6
Thanks for clarifying! :)
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It studies the properties and operations of linear systems, such as matrices, vectors, and linear transformations.

2. What is a matrix?

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It is used to represent and perform calculations on linear equations and transformations.

3. How are linear operators represented by matrices?

A linear operator is a function that maps a vector space onto itself in a linear manner. It can be represented by a matrix by identifying the basis vectors of the vector space and their images under the operator's transformation.

4. What are the applications of linear algebra in science?

Linear algebra has many applications in science, including physics, engineering, computer graphics, and data analysis. It is used to model and solve complex systems, such as quantum mechanics, electrical circuits, and differential equations.

5. How can I learn linear algebra?

There are many resources available for learning linear algebra, such as textbooks, online courses, and tutorials. It is important to have a strong foundation in algebra and calculus before studying linear algebra. Practice and application of the concepts is also key to understanding and mastering the subject.

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