Definition 1.2.1.
A vector space over \(\RR\) is a set \(V\) with addition and scalar multiplication (if \(\alpha \in
\RR\) and \(v \in V\text{,}\) when \(\alpha v\) is defined and remains in \(V\)).
Section 1.2 Linear Algebra
Like much of academic mathematics, piece of linear algebra are necessary for this course. In this section, I’ll go over the most important ideas and definition that I’ll use from linear algebra.
Example 1.2.2.
\(\RR^n\) is a vector space. The scalar multiplication is given by multiplying each component by the real number.
Example 1.2.3.
If \(A\) is \(\RR\) or an interval subset, then \(C(A),
C^n(A)\) and \(C^\infty(A)\) are vector spaces. Scalar multiplication is simply multiplying a constant by a scalar.
Definition 1.2.4.
Let \(V\) be a real vector space and Let \(v_1, \ldots,
v_k \in V\text{.}\) A linear combination of these vectors is a sum \(\alpha_1v_1 + \alpha_2v_2 + \ldots +
\alpha_kv_k\) where \(\alpha_i \in \RR\text{.}\) These vectors are called linearly independent if the equation
\begin{equation*}
\alpha_1 v_1 + \alpha_2 v_2 + \ldots + \alpha_k v_k = 0
\end{equation*}
has only the trivial solution, where all \(\alpha_i =0\text{.}\) Otherwise, the set of vectors is called linearly dependent. A maximal linearly independent set in \(V\) is called a basis (plural bases). The dimension of \(V\) is the number of vector in any bases.
Example 1.2.5.
The vector space \(\RR^n\) has dimension \(n\text{.}\) The vectors spaces \(C(A), C^n(A)\) and \(C^\infty(A)\) are all infinte dimensional.
Definition 1.2.6.
If \(V_1\) and \(V_2\) are vector spaces, a linear transformation \(f:V_1 \rightarrow V_2\) is a function which respects addition and scalar multiplication.
\begin{align*}
f(u+v) \amp = f(u) + f(v)\\
f(\alpha v) \amp = \alpha f(v)
\end{align*}
If \(V_1\) and \(V_2\) are finite dimensional, all linear transformations can be encoded by matrices using matrix multiplication acting on vectors.
Definition 1.2.7.
If \(M\) is a square matrix, then the determinant of \(M\), written \(\det M\text{,}\) is a real number with two properties.
- \(|\det M|\) is the effect of the transformation on the appropriate notion of size (length, area, volume, hypervolume, etc).
- If \(\det M\) is positive, then \(M\) preserves orientation; if \(\det M\) is negative, then \(M\) reverses orientation.
In this course, calculating determinant can simply be done by computer (Wolfram Alpha or other computer algebra system). If you are interested in the calculation of determinant, please consult my linear algebra notes (or any other linear algebra reference) for the details.
Definition 1.2.8.
Let \(f:V_1 \rightarrow V_2\) be a linear transformation. A vector \(v \in V_1\) is an eigenvector for \(f\) with eigenvalue \(\lambda\) if \(f(v) = \lambda
v\text{.}\)
