The definition of vectors.
The matrix with only one column is called a (column)vector, and the matrix with only one row is called a row vector.
Section 2.1 Vector
Subsection 2.1.1 row and column vectors
Some row and column of a matrix can be recorded by
row(i) and column(j)
Example: Let
\begin{equation*}
A = \begin{pmatrix}
0 \amp 2 \amp -8 \amp 8 \\
1 \amp -2 \amp 1 \amp 0 \\
5 \amp 0 \amp -5 \amp 10
\end{pmatrix}
\end{equation*}
Theorem 2.1.1. Vector equation.
Let \(A\) be a matrix of size \(m\times n\text{.}\) Denote the columns of \(A\) as \(\alpha_{1},\alpha_{2},\ldots, \alpha_{n}\text{,}\) that is,
\begin{equation*}
A = [\alpha_{1}\, \alpha_{2}\,\ldots\, \alpha_{n}]\text{.}
\end{equation*}
Then
\begin{equation*}
A\mathbf{x}=x_{1}\alpha_{1}+ x_{2}\alpha_{2}+\ldots+x_{n}\alpha_{n}.
\end{equation*}
Subsection 2.1.1.1 How to get a submatrix
Exercise: For the matrix \(U\) below, get the submatrix \(X\) which from the columns 1,2,5; and get the submatrix \(Y\) which from the columns 1,2 and rows 3, 4.
In this course, deleting one row and one column is important. For your information, it will be used in Chapter 3 to compute the Determinant of a square matrix.
Exercise: Delete the column 3 to form a new matrix \(F\text{.}\)
