Vectors

A vector is a matrix of one column.

Linear Combination

We can describe vectors as the sum of vectors—a linear combination.

\[\vec w = c_1\vec v_1 + c_2\vec v_2 + \cdots + c_n\vec v_n\]

In the same way that we were able to represent a linear system of equations as an augmented matrix, we can represent a system with a linear combination, thinking variable-wise.

\[\begin{align*} & 2x &&+ y &&- z &&= 1 \\ & 3x &&+ 4y &&+ 2z &&= 13 \\ & x &&- 5y &&- 2z &&= 0 \end{align*}\] \[\begin{bmatrix} \begin{array}{ccc|c} 2 & 1 & -1 & 1 \\ 3 & 4 & 2 & 13 \\ 1 & -5 & -2 & 0 \end{array} \end{bmatrix}\] \[x \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} + y \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix} + z \begin{bmatrix} -1 \\ 2 \\ -2 \end{bmatrix} = \begin{bmatrix} 1 \\ 13 \\ 0 \end{bmatrix}\]