> For the complete documentation index, see [llms.txt](https://huang-jason.gitbook.io/math/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://huang-jason.gitbook.io/math/linea-algebra.md).

# linear algebra 1

## Transpose

$$\boldsymbol{(AB)}^T = \boldsymbol{B}^T \cdot \boldsymbol{A}^T$$

![](https://3261671725-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lq83PzGfjVutWtDeb3l%2F-Lq83T8WxyHGIG5BRnUx%2F-Lq83TWAcsXnEzc55Grn%2Fmatrix_transpose.png?generation=1569962722517582\&alt=media)

## Element-wise op, Hadamard product

![](https://3261671725-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lq83PzGfjVutWtDeb3l%2F-Lq83T8WxyHGIG5BRnUx%2F-Lq83TWDeEB-aJwQ9XFL%2Fhadamard.png?generation=1569962715957144\&alt=media)

## Dot product

* projection = Mechanical work is the dot product of force and displacement vectors.
* output is scalar.

![](https://3261671725-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lq83PzGfjVutWtDeb3l%2F-Lq83T8WxyHGIG5BRnUx%2F-Lq83TWFSp_D_vnqe66w%2Fdot_product.png?generation=1569962715884498\&alt=media)

## Matrix multiplication is not commutative.

$$\boldsymbol{A} \cdot \boldsymbol{B} \stackrel{?}{=} \boldsymbol{B} \cdot \boldsymbol{A}$$

## Identity Matrix

![](https://3261671725-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lq83PzGfjVutWtDeb3l%2F-Lq83T8WxyHGIG5BRnUx%2F-Lq83TWHzYWt28bJ41Qo%2Fidentity_matrix.png?generation=1569962715955578\&alt=media)

## Matrix inverse

$$\boldsymbol{A} \cdot \boldsymbol{A} ^{-1} = \boldsymbol{I}$$

For $$\boldsymbol{A} ^ {-1}$$ to be exist, $$\boldsymbol{Ax} = \boldsymbol{b}$$ must have exactly one solution.

Necessary & sufficient conditions to find inverse matrix = **A** is square, **m = n = rank(A)**

## Rank

* linearly independent columns
* vector space generated by its columns

## Norms

* mapping vectors to non-negative values.

![](https://3261671725-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lq83PzGfjVutWtDeb3l%2F-Lq83T8WxyHGIG5BRnUx%2F-Lq83TWJ2JxBKqcRzYcK%2Flp_norm.png?generation=1569962716115394\&alt=media)

$$L ^ 2$$ norm, $$||x||$$ = Euclidean distance

squared $$L ^ 2$$norm = better computationally, but increase slowly near the origin.

$$L ^ 1$$norm = good for zero & nonzero is important.

$$L ^ \infty$$norm = absolute value of the element with the largest magnitude in the vector.