Matrix-vector products as functions

• In calc we play with functions $f:R\to R$, that map a domain of $R$ onto a codomain of $R$.
• Here we have functions $f_A:R^n\to R^m$ that map vectors not just real numbers.
• In fact, matrix-vector product almost functions like a function. Given a vector $\vec{x}$, it returns only one $\vec{y}$. So it passes the vertical line test.
• This basically motivates us to actually define a matrix transformation.

Matrix Transformation

• We have a function $f_A:R^n\to R^m$

• It’s defined such: $f_A(\vec{x})=A\vec{x}$ for every $\vec{x}\in R^n$

• So $f_A$ is the matrix transformation corresponding to $A$.

• $A\vec{x}$ is the image of $\vec{x}$ under $f_A$.

• It’s basically a function that transforms a vector using a matrix-vector product.