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coursera week 1 - matrices and vectors

1. Matrix Elements

$ A_{ij} = $ “$i, j$ entry” in the $i^{th}$ row, $j^{th}$ column

$ A_{32} = 8 $

2. Vector $A_n$ n*1 matrix

$R^4$ 4 dimensional vector
$y_i = i^{th} element$

2.1 math 1-indexed

2.2 machine-learning 0-indexed

3. Matrix Addition

4. Scalar Multiplication

5. Combination of Operands

6. Matrix Vector Multiplication

Matrix Vector Multiplication Fmt :

Matrix Vector

6.1 House sizes example

House sizes Price
2104 ?
1416 ?
1534 ?
852 ?

7. Practice Example

$ A_{2 \times 3} \times A_{3 \times 2} = A_{2 \times 2} $

Matrix

8. House Example

Matrix

9. Matrix $A \times B \neq B \times A$

But, 结合律,可以的

$ A \times B \times C = (A \times B) \times C = A \times (B \times C) $

10. Identity Matrix

Denoted I (or I_{n*n}).

10.1 $2 \times 2$

10.2 $3 \times 3$

$Z \times I = I \times Z = Z$

11. Matrix Inverse

$3 \times 3^{-1} = 1$

Not all numbers have an inverse.

if $A$ is an $m \times m$ matrix, and if it has an inverse

$A \times A^{-1} = A^{-1} \times A = I$

$ A \times A^{-1} = I_{2 \times 2} $

12. Matrix Transpose

Let $A$ be an $m \times n$ matrix, and let $B = A^T$.
Then $B$ is an $n \times m$ matrix, and $B_{ij} = A_{ji}$