The central problem of linear algebra is to solve linear equations.
- System: 2 or more equations
- Linear: unknowns are multiplied by numbers
- Equation: Value of 2 mathematical expressions are equal
- Can I solve every Ax = b for every b?
- Does the Linear Combination of the columns fill the n-space? If columns are independent and A is not singular → yes, else no.
Not singular = the inverse exists.
#Types of Solutions
- Unique solution
- Infinite solutions
- No solution
#2X2 Row Picture
#2X2 Column Picture
#2X2 Matrix Picture
#3X3 Row Picture
#3X3 Column Picture
#3X3 Matrix Picture
Example:
$$ \begin{cases} x + 2y + 3z = 6 \\ 2x + 5y + 2z = 4 \\ 6x - 3y + z = 2 \end{cases} $$ $$ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 2 \\ 6 & -3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ 2 \end{bmatrix} $$ $$ A \cdot x = b $$
Syllabus