1-Matrix Interpretation

Interpretation of equation sets¶

• Suppose we have the following equation sets

Row Picture¶

• Hard to draw when dimensions goes beyond 2

Column Picture¶

• Linear Combination of Columns
  

Quiz¶

Elimination¶

• A systematic way to solve the solution

Upper triangular matrix¶

The most important purpose in elimination is from $A→U$(upper triangular matrix )

Augmented matrix¶

• Add y (response) into the x's (parameters') matrix

Failure of Elimination¶

• Some pivot become 0, which means there are some vectors are parallel(dependent) to each other

Elimination matrix¶

• Use matrix to represent the elimination steps

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• $E_{21}$ means elimination row 1 using row 2

Permutation matrix $P$¶

• Exchange the row

Inverse Matrix¶

• How we can reversing steps and transfer matrix from $U \to A$? (Inverse Matrix)
  

Quiz¶

Yalin Conclusion¶

Row interpretation is suitable for first matrix, column interpretation is for second matrix [行前列后]¶

• $E_{row operation}*A=U$
• $A*E_{column operation} = U$

The meaning of Matrix pre-multiplication(矩阵左乘)¶