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Course Notes for Multivariable Calculus

Remkes Kooistra

Section 4.4 Matrices and Linear Transformations

Subsection 4.4.1 Linear Transformations

This section is a short detour into a few definitions from linear algebra. This is necessary to state the important results in the section immediately following.
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Definition 4.4.1.

Let \(L\) be a function \(\RR^n \rightarrow \RR^m\text{.}\) Then \(L\) is called a linear transformation if the follow two properties hold for all \(u, v, \in \RR^n\) and \(a \in \RR\text{.}\)
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  1. \(\displaystyle L(u + v) = L(u) + L(v)\)
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  2. \(\displaystyle L(au) = aL(u)\)
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Linear transformations are the functions which are studies in linear algebra. They are a very restricted class of function: most expression to not preserve addition and scalar multiplication in this way. Note that by using \(a=0\) in the second property, a linear transformation must always send the zero vector to the zero vector
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As limited as they are, linear transformations are a key definition in algebra and geometry. They also have a very elegant catalogue and calculation method, which involves matrices. Very briefly, let me define a matrix.
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Definition 4.4.2.

A matrix is a rectangular array of numbers. It has a number of rows and columns; if there are \(n\) rows and \(m\) columms, then it is called a \(n \times m\) matrix. If the number of rows and columns are the same, it is called a square matrix.
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Here is an example of a \(3 \times 3\) matrix where the entries are real number.
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\begin{equation*} \begin{pmatrix} \sqrt{7} \amp 0 \amp \frac{1}{e} \\ \frac{-15}{31} \amp \pi \amp 5 \\ 0 \amp 0 \amp 0 \end{pmatrix} \end{equation*}
Matrices are used for a huge number of purposes in mathematics. For this course, we will only need a couple. The first is one of the most notable and surprising facts in mathematics.
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How is this done? It is done by the matrix action on vectors: there is a clear and algorithmic way that a matrix can be used to take a vector as input and produce another vector as output. I’ll state the matrix action just in \(\RR^3\) for square matrices, but it applies to any size of matrix.
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Definition 4.4.4.

Let \(M\) be a \(3 \times 3\) matrix and \((x,y,z)\) be a vector in \(\RR^3\text{.}\) Then, writing the matrix out as nine entries, the following formula defines the matrix action. (Usually, the vector would be written vertically for this action, but I’ll write it horizontally to conform to the notation of this course. That’ll look a bit wierd to those who have done linear algebra with me -- sorry).
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\begin{equation*} \begin{pmatrix} a \amp b \amp c \\ d \amp e \amp f \\ g \amp h \amp i \end{pmatrix} (x,y,z) = (ax + by + cz, de + ey + fz, gx + hy + iz) \end{equation*}
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The details of calculating the matrix action won’t actually be that important for us, but I do want you to know that the action exists. For now, there is a second property of matrices which we will need.
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Definition 4.4.5.

Let \(M\) be a square matrix. The there is a real number, called the determinant of M and written \(\det(M)\text{,}\) with two properties: the absolute value \(|\det(M)|\) is the multiplicative effect that \(M\) has as a linear tranformation on volume (or the appropriate equivalent in other dimension); and the sign of the determinant records the effect of \(M\) on orientation, with positive preserving orientation and negative reversing orientation.
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Again, the details here won’t be extremely important for us, so if you don’t know exactly what I mean by multiplicative effect on volume or what orientiation is, that’s fine. It’s enough to know that this determinant exists and carries important geometric meaning. Likewise, calculating determinants is also not going to be required: I will either explicilty give you the determinant calculations or such calculations can be done by computer.
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