Review and Outlook
Let’s review linear combinations in 2D space from a view of the inner product.
\[ \textbf{y} = x_1 \cdot \textbf{a}_1 + x_2 \cdot \textbf{a}_2 \]
The vector \(\textbf{y}\) is represented as the linear combination of the two basis vectors \(\textbf{a}_1, \textbf{a}_2\). It can be reviewed as the inner product of two “vectors”, one is the vector of coefficients, \((x_1, x_2)^{\top}\), that we are familiar with, and another is a vector of two vectors, \((\textbf{a}_1, \textbf{a}_2)^{\top}\). Based on this thinking, we can rewrite it as
\[ \textbf{y} = (\textbf{a}_1, \textbf{a}_2) \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \]
Let’s focus on \((\textbf{a}_1, \textbf{a}_2)\). It is a 2 by 2 rectangle array and we denote it by a bold capital letter, \(\textbf{A}\), and name it matrix!
