Linear Albegra Lecture 1 - Geometry of Linear Equations

Linear Equations vs line and pland vs vectors.

Linear equations:

$$2x - y = 0 \\
-x + 2y = 3$$

Write in matrix, we have：

$$ \begin{bmatrix} 2 & -1 \\
-1 & 2y \end{bmatrix} \begin{bmatrix} x \\
y \end{bmatrix} = \begin{bmatrix} 0 \\
3 \end{bmatrix} $$

or in short $$ A X = b $$ A is called the matrix, X is the unknown.

The tow picture：

row is presents a line and the joint point of line is the solution of the linear equations.

$$ x \begin{bmatrix} 2 \\
-1 \end{bmatrix} + y \begin{bmatrix} -1 \\
-2 \end{bmatrix} = \begin{bmatrix} 0 \\
3 \end{bmatrix} $$

This is linear combination of columns, which represents adding two vectors and get the result of third vector.

Column picture:

In 3D, row picture represents a plane.

3D column picture is three vectors added up to be the fourth vector b.

Think about the cases we do no have a solutions.