The Matrix of a Linear Transformation

[clever cape]

Hello, I'm a bit tripped up on the notation. In the previous section, a linear transformation T(x) is equal to matrix equation Ax, where A is what transforms x into a new vector b (Ax=b)

I don't quite understand whats going on with the notation. They set it up in a vector equation x1e1 + x2e2=X and it looks like the matrix equation is meant to be Ix, where I looks like its a matrix A.

The previous section always described the transformation to be what happens to the inputs X, so why is it that in this section the transformation is occurring to the columns of the matrix instead of the inputs X?

[tepid moon]

this comes from the linearity of linear/matrix transformations
so first is it mentioned that $T(ax+by) = aT(x) + bT(y)$ for any constants $a,b$ and vectors $x,y$?

[carmine cobalt]

Dee3Cay

[clever cape]

But the problem im having is that A(u+v)=Au+Av=T(u+v)=b. This is of the form Ax. In the example, A is the identity matrix I, giving us the matrix equation Ax=>Ix, where A=I (im not sure if this is sound reasoning)

So the linear transformation should be T(x)=Ix instead of T(e1) + T(e2) since e1 and e2 are coming from the matrix I? Im not having problems with the linearity of a matrix, im just not seeing how this notation connects with vector and matrix equations

[tepid moon]

"But the problem im having is that A(u+v)=Au+Av=T(u+v)=b." ok so you understand this part

now let's see what's happening

first, do you understand this part

[clever cape]

gotcha 👍 practically saying the same thing as Ax=[a1x1 a2x2 .... an*xn]

ahhhh i think im seeing it now

Ix=e1x1 + e2x2 + ... + enxn=b= T(e1.....en)

[tepid moon]

yee

fyr

[clever cape]

and then from here we know that e1=[1, 0] and e2=[0, 1]

T(x1e1)+T(x2e2) and x1 & x2 might as well be weights or constants that we can pull out where we would get x1T(e1)+x2T(e2)

[tepid moon]

yes

this idea shows that not only can you use the x = x1e1 + x2e2 sort of linear combinations, but also T(x) = x1T(e1) + x2T(e2)

so the linear combination is somehow "preserved" after transformation

[clever cape]

Gotcha! I appreciate the help, thank you very much!!

How do I mark this as solved?

[tepid moon]

idk tbh lmao