# Do I need to understand Multi-Variable Calculus to study Linear Algebra?

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?

• Listen to Dr. Strang's OCW lectures and see if you can follow them. I don't think you necessarily need multivariate calculus but being exposed to higher level maths coming into Linear Algebra will definitely help. – mathmath8128 Sep 19 '11 at 1:48
• You need linear algebra to understand multi-variable calculus, not the other way around. – Yuri Sulyma Sep 19 '11 at 1:51

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.

You don't need to understand multi-Variable calculus to study linear algebra. In fact I think linear algebra would help for you to understand multi-Variable calculus.

• That seems to be the sentiment. I guess I'll be giving linear algebra a go! – thompsonjames Sep 19 '11 at 12:37

The two are almost completely independent. In linear algebra terms, they are orthogonal to each other. In linear algebra you're learning techniques and rules for matrix manipulation, whereas in multi-variable calculus you'll be learning about partial differentials and how to integrate over multiple variables. Nothing in common except maybe coincidental overlaps of minor topics.

• -1: I disagree with this answer. The Jacobian and the matrix of partials describing a tangent plane (or a function's local linear approximation) are two serious examples of linear algebraic concepts in multivariable calculus. (And, no offense, but I also have this extra $1$ in my reputation modulo $5$ that's been bothering me.) – anon Sep 19 '11 at 7:36
• This is exactly what I mean by coincidental overlap. That's what, 1 or 2 days out of a 90 day calculus class? – Doug Treadwell Sep 19 '11 at 9:15
• I think change-of-coordinates is a pretty significant feature of multivariable calculus (e.g. can't evaluate volume content of curvilinear images without it, or indeed surface area without an implicit determinant), frequently recurring after it's introduced. Also relevant and more elementary are linear independence and orthogonality, rotations and affine transformations, solving for lines and intersections, projections and subspaces, plus segues into higher concepts like curvature and tangent bundles. Vector spaces naturally coincide with calculus through the geometric substance. – anon Sep 19 '11 at 10:47
• I think you're confusing lower division multi-variable calculus as it's taught with how you could theoretically apply it or how it is used in later courses. Knowledge of linear algebra or multi-variable calculus will not greatly effect the ease of taking the other. In the schools I looked at, multi-variable calculus is not a prerequisite for linear algebra or vice versa, which indicates that what I'm saying is correct. – Doug Treadwell Sep 19 '11 at 18:59
• Naw, I actually took a course in Calc III (and nothing beyond it) that included everything I mentioned straight out of the book (except bundles, of course). The thing is that only a handful of significant linear-algebraic concepts are directly relevant (vector spaces, linear maps, solving a system), and they are specifically useful in the basic vector algebra problems (also e.g. bases, spaces, span, dimension), so learning the necessary L.A. gets subsumed into the introductory material quite easily despite how its effects are felt throughout the course. – anon Sep 19 '11 at 19:41