What is the Linear Transformation for this Matrix? $$
A=\pmatrix{3&5&-4\\-3&-2&4\\6&1&-8}\pmatrix{x1\\x2\\x3}=\pmatrix{b1\\b2\\b3}
$$
Given the above matrix A, there exists a linear transformation known as "T", what is this transformation and it goes to and from what space?
 A: What is usually meant by the linear transformation $T$ described by matrix $A$ can be described as follows:
For a (column-)vector $x$ of the correct size (in this case, $x$ must have $3$ entries), we say that $T(x) = Ax$.  That is, $T(x)$ is what you get when you matrix-multiply $A$ by $x$.
To write this out in its gory detail: denote $x = (x_1,x_2,x_3)^T$.  We then have
$$
T(x) = Ax = 
\pmatrix{3&5&-4\\-3&-2&4\\6&1&-8} \pmatrix{x_1\\x_2\\x_3} =
\pmatrix{
3x_1 + 5x_2 - 4x_3\\
-3x_1 + 2x_2 + 4x_3\\
6x_1 + x_2 - 8x_3
}
$$
Since $T$ takes in a vector of $3$ entries and yields a vector of $3$ entries, it goes from the space $\Bbb R^3$ to the space $\Bbb R^3$.
In general, the number of columns of $A$ is the dimension of the domain ("starting space") of $T$ and the number of rows is the dimension of the codomain ("destination space") of $T$.
A: I'm guessing it wants you to define a linear transformation $T$ using this $A$. Let's pick a simple one,
$$ T: \mathbb{R}^3 \to \mathbb{R}^3 \quad by \quad T(x)=Ax $$
