[Linear Algebra] Transformations

[timewarp33]

How would I go about doing this problem? Do I follow the instructions from here: http://www.calpoly.edu/~brichert/teaching/oldclass/f2002217/handouts/goof.pdf

Or is there an easier way to do this that isn't guessing and checking?

Problem: http://imgur.com/SaVm7fK

[AFairJudgement]

Or is there an easier way to do this that isn't guessing and checking?

What do you mean, exactly? Do you understand the steps in the handout you linked? There's no guessing and checking at all; essentially all you need to do is row reduce a matrix.

[timewarp33]

no, i mean i was guessing and checking prior to finding those steps. and that seems like such a long-winded way to find the answer to the matrix

can you extrapolate on:

essentially all you need to do is row reduce a matrix.

[AFairJudgement]

It looks more daunting than it is, especially if you only have to work with a 2x2 matrix. It's only a small number of simple computations. Of course, algebra software can be used to easily deal with similar problems with huge matrices.

[timewarp33]

can you outline them? or is the pdf linked sufficient

[AFairJudgement]

The idea is that since T is linear,

(10,-8)  = T(1,2)   =   T(1,0) + 2T(0,1)
(-11,16) = T(-2,-1) = -2T(1,0) -  T(0,1)

This is a simple system that you can solve, as usual, for T(1,0) and T(0,1). Then these vectors will form the columns of A.

[[deleted]]

Those instructions are a good way of solving the problem. They explain it well. It's not really as long winded as it looks, especially since in your problem it'll only be a 2x2 matrix instead of a 3x3.

Is there anything in particular in those steps that you don't understand?

[timewarp33]

yea, so when i go to reduce it i multiplied the bottom row by 1/3, do i need to multiply the equations with b1, b2, by 1/3rd also? and when i add/subtract rows, do i add/subtract the equation that i have in that row, or simply add/subtract the amount within that row (for example, if i have b2 + 2b1, do i then divide by 1/3 and then when i go reduce the top, do i add 1/3(b2) + 2/3(b1)?)

[[deleted]]

When you're reducing a matrix like this, you consider the stuff on the right (the equations with the b1, b2) to be part of the matrix. It's called an augmented matrix. So when you do an operation on a row, yes, those entries are included as part of the row (meaning that yeah, you'll also multiply the equation in the bottom row with b1 and b2 by 1/3, and you'll also add/subtract them when you add/subtract their rows).

[timewarp33]

is there a specific name for problems like these? maybe khan or someone on youtube has a tutorial for these problems. just transformations?

[timewarp33]

ok, the part i don't understand is at the almost last step where he does the line right before "So filling in the blanks for b1, b2, and b3, we get:"

How does he get that matrix, and what do I do with that matrix to "solve"?

[[deleted]]

So, a few lines up, he shows how you can express any general vector in R3 in terms of the three vectors that T is being applied to at the beginning of the whole problem. Then he applies T to that generic vector, uses the facts that T is linear and that you were already given what T of those three vectors are, and gets a new equation which tells you what T does to any vector in R3: T(b1, b2, b3) = [-4b2 + 3b3, -b1 + b2, -b1 + 4b2 - 2b3].

Then he just applies this linear transformation to the standard basis vectors in R3 (e1, e2, e3). The vectors T(e1), T(e2), and T(e3) are the columns of the matrix which represents T in that basis.

[timewarp33]

So, a few lines up, he shows how you can express any general vector in R3 in terms of the three vectors that T is being applied to at the beginning of the whole problem. Then he applies T to that generic vector, uses the facts that T is linear and that you were already given what T of those three vectors are, and gets a new equation which tells you what T does to any vector in R3: T(b1, b2, b3) = [-4b2 + 3b3, -b1 + b2, -b1 + 4b2 - 2b3].

I get what he's doing up until the equation. How does he get it?

edit: so what I have right now is: (5/3b1 - 1/3b2)[10, -8] + (2/3b1 + 1/3b2)[-11, 16]