I am currently taking a intro course to abstract algebra and am revisiting ideas from linear algebra so that I can better understand examples.
When i was in undergraduate learning L.A., I thought of matrix manipulations as ways of solving $n \times n$ systems of equations. Recently i was exposed to the idea of a matrix being a linear transformation, and matrix multiplication being composition of linear transformations. Im trying to understand this in a more intuitive way and was hoping for some insight…
I was thinking of a basic $2\times2$ example and how it affects a point $(x,y)$.
We could have a matrix :
\begin{bmatrix} a & b \\ c & d \end{bmatrix}
When we 'apply' or multiply this to a point $(x,y)$ using matrix multiplication we get new $x' = ax + by$ and $y' = cx + dy$.
So if $b,c = 0$, then I can see that what we are doing is 'scaling' both $x \;\& \;y$. I'm guessing that if $b,c \neq 0$, then this becomes some sort of rotation or reflection, but how do you understand this on a fundamental level?
How do these operations relate to Gaussian elimination when we are trying to solve systems of equations? Or are are these two seperate applications of matrices?
Another observation is that when multiplying a matrix such as this one with a point, we get two equations which remind me of Bézout's identity. Am I overanalyzing this or can I draw connections between these two concepts?
Thanks for any input!
Best Answer
Personally, I don't feel that Gaussian elimination is the best ways to view linear transformations. It is a standard theorem that every linear mapping can be represented as a matrix under a certain basis. For simplicity, I will stick with the standard basis.
Suppose we have a linear mapping $T:\ \mathbb{F}^n \rightarrow \mathbb{F}^n$. The action of the linear mapping is entirely determined by it's action on the basis vectors. If we let $$\mathbf{v} = c_1\mathbf{e_1} + \cdots + c_n\mathbf{e_n}$$ then correspondingly $$T(\mathbf{v}) = c_1T(\mathbf{e_1}) + \cdots + c_nT(\mathbf{e_n})$$ so that knowing the set $\left\{T(\mathbf{e_1}),\ \cdots,\ T(\mathbf{e_n})\right\}$ is enough to determine the nature of the mapping.
Conversely, this property allows show to determine how a matrix acts as a mapping. If we have a matrix $$A = \begin{pmatrix}\mathbf{a_1} & \cdots & \mathbf{a_n}\end{pmatrix}$$ where $\mathbf{a_i}$ are the column vectors of $A$, then the action of the mapping induced by multiplication by $A$ will be $$A\mathbf{e_i} = \mathbf{a_i}$$ Multiplying $A$ by the $i$th standard basis vector has the effect of selecting the $i$th column of $A$. You can clearly see the action of the mapping in this way; the matrix maps the $i$th standard basis vector to it's $i$th column.
These methods give a geometric interpretation to the matrices, but it is not necessarily the most natural one. Just because we know the action of the mapping doesn't mean that we necessarily understand it. This is why the concept of diagonalization (or more generally Jordanization as tomasz mentions) is introduced. Diagonalization is the process of selecting the most geometrically natural basis for the mapping.
If the basis $\left\{\mathbf{v_i}, \cdots,\ \mathbf{v_n}\right\}$ diagonalizes $A$ to $$D = \mathrm{diag}(\lambda_1,\ \cdots,\ \lambda_n)$$ then the action of the mapping is a dilation of factor $\lambda_i$ in the direction of the $i$th basis vector, $\mathbf{v_i}$.