Linear combinations pictures

For vector u, v, w

Sometimes, 3 vectors fill a plane instead of a space, criteria as follows --

For vector u, v, w, if you can find c, d so that w = cu + dv, then w lies on the plane of cu + dv, and u+v+w also forms the plane of cu + dv

Solving linear equation

Elementary Row Operations

Example

Linear Equation Solution Pattern

Possible consequences

Non-unique solutions

Homogeneous systems

• 等号后都是0.
• Consistent -- 永远有解（x1=x2=x3=...=xn=0) -- trivial solution

**如果m>=n, 矩阵有unique solution (trivial) or there are infinitely many nontrivial solutions

Matrix Operation Properties

• 加减法满足交换律，结合律        
•  乘除法满足交换律，结合律      
• 混合运算

 

 Transpose

Rows become columns, columns become rows, denoted as 

Symmetric Matrix

 

 OR

symmetric across the main diagonal

Example

Inner & Outer Product