What does it mean when someone says \"T is a linear transformationfrom R m to R

Question

What does it mean when someone says "T is a linear transformationfrom Rm to Rn ?
for example:

T(x)=Ax, if T is from Rm to Rn what does itmean? how does it apply to rows and columns of the matrix?

Explanation / Answer

If T : R^m -> R^n is a linear transformation means, that thevector from the domain is in R^m and is mapped into R^n. The way to picture this with matrix multiplication, is exactly asyou wrote, Ax where x is a column vector. This implies that x has mrows and one column, and in order for the multiplication to be welldefined, A should have m columns. Because you want to map it intoR^n, this means that you should have n rows in the matrix A, so tohave n numbers for the new mapped vector. This should not only reveal that A has m columns and n rows, butthat the content of each row of A is the way you write then-component of the new vector as a linear combination of thecomponents of the vector x.

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