I'm currently studying Single-Variable Calculus independantly through MIT OCW. I can only focus on one course at a time independantly since it takes up so much time, and I really want to study Linear Algebra next instead of Multi-Variable Calculus.
My question is simply this: after understanding Single-Variable Calculus, would I be able to continue into and understand Linear Algebra, or should I do Multi-Variable Calculus first?
Best Answer
Multivariable calculus is helpful because it gives many applications of linear algebra, but it's certainly not necessary. In fact, you probably need linear algebra to really start to understand multivariable calculus.
To wit, one of the central objects in multivariable calculus is the differential of a function. In single-variable calculus, you are taught that the differential of a function $f:\mathbb{R}\to\mathbb{R}$ is a new map $f':\mathbb{R}\to\mathbb{R}$ which provides the slope of the tangent line to $f$ at each point in $\mathbb{R}$. This is strictly correct, but it is not the best way to understand single-variable calculus if you want to easily generalize.
The better way to see single-variable calculus is to recall that the tangent line to $f$ at $x$ is the best affine-linear approximation to $f$ at $x$, i.e., $f$ is approximated by $f(y)\approx f'(x)(y - x) + f(x).$
This generalizes quite well! If $f:\mathbb{R}^n\to\mathbb{R}^m$, the differential to $f$ at $x$, $df_x$, is the best linear approximation to $f$ at $x$: $f(y)\approx df_x(y-x) + f(x)$. Now, we think of $x$ and $y$ as vectors in $\mathbb{R}^n$ and the differential $df_x$ is an $n\times m$ matrix.
Even more generally, we think of $df$ as a map from $\mathbb{R}^n$ into $Hom(\mathbb{R}^n,\mathbb{R}^m)$ which measures the best linear approximation of $f$ at each point $x\in\mathbb{R}^n$.
Generalizing further requires the notion, from differential geometry, of a smooth manifold. Such manifolds carry objects called tangent bundles, which assign to each point of the manifold an abstract vector space.
You can see how linear algebra is a little more helpful for multivariable calculus than the other way around.