Matrix Multiplication and Function Composition

[Septimra]

I am doing linear algebra and want to fully understand it, not just pass the class. I was recently taught matrix multiplication and decided to look up how it works. The good part is that I understand the concept. Matrices are a way of representing linear transformations. So matrix multiplication is actually a composition of functions. That is why it is not communicative and it is associative.

But i recently came across this article and I could not follow the math near the middle of the page.
http://nolaymanleftbehind.wordpress.com/2011/07/10/linear-algebra-what-matrices-actually-are/

the matrices that are being multiplied are

[ 2 1 ] [ 1 2 ]
[ 4 3 ] [ 1 0 ]

the basis are w1 and w 2

and w1 = [ 1 0 ]
and w2 = [ 0 1 ]

The author states that all that is needed it to see how the linear transformation affects the basis vectors.

Then it states that f(g(w1)) = f(w1+w2)
How does that work? Where on Earth do you plug in the w1?
Please help

 

[Simon Bridge]

That would depend on what the author sees as f and g ... but the basic principle is that a linear transformation can be represented as a transformation of the coordinate system. A square in an oblique coordinate system looks the same as an oblique shape in a rectangular coordinate system.

 

[Fredrik]

But i recently came across this article and I could not follow the math near the middle of the page.
http://nolaymanleftbehind.wordpress.com/2011/07/10/linear-algebra-what-matrices-actually-are/

the matrices that are being multiplied are

[ 2 1 ] [ 1 2 ]
[ 4 3 ] [ 1 0 ]

the basis are w1 and w 2

and w1 = [ 1 0 ]
and w2 = [ 0 1 ]

The author states that all that is needed it to see how the linear transformation affects the basis vectors.

Then it states that f(g(w1)) = f(w1+w2)
How does that work? Where on Earth do you plug in the w1?
Please help

If we write elements of $\mathbb R^2$ as 2×1 matrices, the definition of $g:\mathbb R^2\to\mathbb R^2$ can be written as $g(x)=Bx$ for all $x\in\mathbb R^2$. So
$$g(w_1)=Bw_1 =\begin{pmatrix}1 & 2\\ 1 & 0\end{pmatrix}\begin{pmatrix}1 \\ 0\end{pmatrix}=\begin{pmatrix}1\\ 1\end{pmatrix}=\begin{pmatrix}1\\ 0\end{pmatrix}+\begin{pmatrix}0\\ 1\end{pmatrix}=w_1+w_2.$$
You may find https://www.physicsforums.com/showthread.php?p=4402648#post4402648 useful.

 

[Septimra]

Thank you a lot, I appreciate it. I now see what the author was saying.
But I still have one minor question. I thought the author was trying to prove that g(x) = Bx. I now see that I was mistaken. But could one of you prove this? How does g(x) = Bx if x is a vector?

 

[Fredrik]

How does g(x) = Bx if x is a vector?

x is an element of $\mathbb R^2$. If we use the convention to write elements of $\mathbb R^2$ as 2×1 matrices, then we can just define $g(x)=Bx$ for all $x\in\mathbb R^2$. If we instead use the convention to write elements of $\mathbb R^2$ in the standard $(x_1,x_2)$ notation for ordered pairs, the notation $Bx$ doesn't work, but we could e.g. define
$$g(x)=\left(\left(B\begin{pmatrix}x_1\\ x_2\end{pmatrix}\right)_1,\left(B\begin{pmatrix}x_1\\ x_2\end{pmatrix}\right)_2\right)$$ for all $x\in\mathbb R^2$. This looks really awkward of course. This is why I chose to use the matrix notation instead of the ordered pair notation.

We could also define g by saying that it's the function defined by $g(s,t)=(s+2t,s)$ for all $s,t\in\mathbb R$. The matrix of this function with respect to the standard ordered basis $(e_1,e_2)$ where $e_1=(1,0)$ and $e_2=(0,1)$, has $g(e_j)_i$ on row i, column j, as explained in the FAQ post. This is the ith component of the vector we get when g takes e_j as input. For example, row 2, column 1, of this matrix is
$$g(e_1)_2=(g(1,0))_2=(1+2\cdot 0,1)_2=1.$$ Note that this is equal to $B_{21}$, as it's supposed to be.

If you want to understand how matrix multiplication is really composition of linear functions, then you should study the FAQ post and do this exercise: Let A and B be linear functions from $\mathbb R^n$ to $\mathbb R^n$. Let [A] and denote their matrix representations with respect to the standard basis for $\mathbb R^n$. Let [AB] denote the matrix representation of AB with respect to the standard basis for $\mathbb R^n$. Prove that for all $i,j\in\{1,\dots,n\}$, we have
$$[A\circ B]_{ij}=[AB]_{ij}.$$ This result tells us that the matrix representation of $A\circ B$ is equal to the matrix product of the matrix representations of A and B.

Hint: The definition of matrix multiplication is $(XY)_{ij}=\sum_k X_{ik}Y_{kj}$. You will also have to use the fact that every vector is a linear combination of basis vectors.