Date modified: Wed 2025-02-19 . 04:07 PM
Date created: Wed 2025-02-19 . 03:53 PM
# Some introductory words. Just some temporary words here. Broadly speaking, linear algebra is the discipline in math that studies the following things: (1) vectorspaces and (2) linear transformations. But what even are these things? As it turns out, you may already know many examples of vectorspaces. For example the set of vectors on the 2-dimensional plane $\mathbf R^{2}$ is an example of a vectorspace. So is the 3-dimensional space $\mathbf R^{3}$. Or the collection of all continuous functions on the real line, denote as $\mathscr C^{0}$, or the collection of functions whose derivative are continuous, denote as $\mathscr C^{1}$, or functions that can be differentiated infinitely often, denote as $\mathscr C^{\infty}$, etc. And linear transformations? These are just functions $T$ between vectorspaces that "preserve linear combination", where $$ T(ax + by)=aT(x)+bT(y), $$where $a,b$ some scalars, and $x,y$ some vectors. But secretly you know some of them already, for example the derivative operator $\frac{d}{dx}$ on functions that can be differentiated infinitely often is a linear transformation! For example, $$ \frac{d}{dx}(3 x^{2} -5\cos(x)) = 3 \frac{d}{dx}(x^{2}) -5 \frac{d}{dx}(\cos(x)). $$ And as we will see, linear transformations between spaces (vectorspaces) like $\mathbf R^{n}$ are just **matrices** of the right shape. A matrix is "just" a **rectangular table of numbers**, something that looks like this $$ \begin{bmatrix}1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12\end{bmatrix}. $$If we know what we are doing, we can do many things with it! It's ok if those words don't quite make sense yet. In this course, we will try to do the following four things: - Develop the theory of linear equations and matrices. - Develop the theory of vectorspaces and linear transformation. - Study the eigenvalue, eigenvector, and diagonalization problem. - Study inner product spaces, and orthogonality. By "theory" we really also mean the **language** behind all of this, so learning what words mean is important. I will try to give applications to these things as we come across them. Ok, but before start, let me give you a taste of what linear algebra can do, something called **singular value decomposition (SVD)**, and how we can use it to **compress an image file**.