Question

Given that T is a linear transformation, derive a formula for T.

Answer 1

Joe, if you have to go, I will understand :-).

Answer 2

bookmark

Answer 3

I need to think about this a little lol. I can see a way to solve the problem, but there might be a shorter way.

Answer 4

I just don't get how to solve these ridiculously hard questions. I would rather be doing diffEq than this!

Answer 5

First, do you notice that the vectors (1,0,1), (1,1,0), and (0,1,1) are linearly independent and form a basis?

Answer 6

Yes.

Answer 7

Good, thats whats going to make this easy. Im going to type out the idea and post it, one sec.

Answer 8

oops oops oops, i was typing too fast. I dont mean the columns of A are linear independent, i meant the columns of that matrix next to A.

Answer 9

Oh my gosh, why did I not think of that?! I should've known to invert that sucker!

Answer 10

Here is the correction.

Answer 11

You're a much better explainer than the book is!

Answer 12

i have to run. Im sure there is probably a shorter or more interesting solution. anyways, have a good day :)

Answer 13

Thanks you SO much, I really mean it.

Answer 14

I passed linear algebra because of joe

Answer 15

Joe is the man!

Answer 16

ok so since we have a linear transformation, we then have an induced matrix A such that
T(x)=Ax
now since our output is an 2x1 matrix and that the vector x we input is an 3x1 matrix, then our induced matrix A is just a 2x3 matrix cause multiplying a 2x3 matrix with a 3x1 matrix will give us an output of 2x1 matrix. So if we find this matrix A, we could now find a formula :D. ok so let our matrix A be:
\[A=\left[\begin{matrix}a & b & c\\ d & e & f\end{matrix}\right]\]
now we'll get:

Answer 17

I just wrote the solution :)) its hard to type lots of matrices

Answer 18

Thank you very much for your solution. But might I ask why you didn't want to just take the inverse since the 3 vectors form a basis?

Answer 19

because I won't get the induced matrix or any formula for T from that :D which means I won't get any answer from that

Answer 20

anyways goodluck with linear algebra which is used in differential equations also and has lots of implications on engineering, physics, economics, biology, chemistry, and math itself.

Answer 21

I did the inverse on both sides and came up with the same answer as what you got. I think it's easy to find the formula if one of the matrices is square. If one of them is not square, then we use the longer method, which is doing an induced matrix and solving for a-f.

Thank you for showing me the other way so that I can use this method for non-square matrices.

Answer 22

oh okay :D