### Linear Algebra

- mathematics of data: multivariate, least square, variance, covariance, PCA
- equotion: y = , where A is a matrix, b is a vector of depency variable
- application in ML
- Dataset and Data Files
- Images and Photographs
- One Hot Encoding: A one hot encoding is a representation of categorical variables as binary vectors.
`encoded = to_categorical(data)`

- Linear Regression. L1 and L2
- Regularization
- Principal Component Analysis. PCA
- Singular-Value Decomposition. SVD. M=U*S*V
- Latent Semantic Analysis. LSA typically, we use
`tf-idf`

rather than number of terms. Through SVD, we know the different docments with same topic or the different terms with same topic - Recommender Systems.
- Deep Learning

### Numpy

- array broadcasting
- add a scalar or one dimension matrix to another matrix. where b is broadcated.
- it oly works when
**when the shape of each dimension in the arrays are equal or one has the dimension size of 1.** - The dimensions are considered in reverse order, starting with the trailing dimension;

### Matrice

- Vector
- lower letter.
- Addtion, Substruction
- Multiplication, Divsion(Same length) a*b or
- Dot product:

- Vector Norm
- Defination: the length of vector
- L1. Manhattan Norm.
`python: norm(vector, 1)`

. Keep coeffiencents of model samll - L2. Euclidean Norm.
`python: norm(vector)`

- Max Norm.
`python: norm(vector, inf)`

- Matrices
- upper letter.
- Addtion, substruction(same dimension)
- Multiplication, Divsion( same dimension)
- Matrix dot product. If , A’s column(n) need to be same size to B’s row(m).
`python: A.dot(B) or A@B`

- Matrix-Vector dot product.
- Matrix-Scalar. element-wise multiplication
- Type of Matrix
- square matrix. m=n. readily to add, mulitpy, rotate
- symmetric matrix.
- triangular matrix.
`python: tril(vector) or triu(vector)`

lower tri or upper tri matrix - Diagonal matrix. only diagonal line has value, doesnot have to be square matrix.
`python: diag(vector)`

- identity matrix. Do not change vector when multiply to it. notatoin as
`python: identity(dimension)`

- orthogonal matrix. Two vectors are orthogonal when dot product is zeor. or . which means the project of to is zero.
**An orthogonal matrix**is a matrix which

- Matrix Operation
- Transpose. number of rows and columns filpped.
`python: A.T`

- Inverse. where
`python: inv(A)`

- Trace. the sum of the values on the main diagonal of matrix.
`python: trace(A)`

- Determinant. a square matrix is a scalar representation of the volume of the matrix. It tell the matrix is invertable. or .
`python: det(A)`

. - Rank. Number of linear indepent row or column(which is less). The number of dimesions spanned by all vectors in the matrix.
`python: rank(A)`

- Transpose. number of rows and columns filpped.
- Sparse matrix
- sparsity score =
- example: word2vector
- space and time complexity
- Data and preperation
- record count of activity: match movie, listen a song, buy a product. It usually be encoded as :
**one hot, count encoding, TF-IDF**

- record count of activity: match movie, listen a song, buy a product. It usually be encoded as :
- Area: NLP, Recomand system, Computer vision with lots of black pixel.
- Solution to represent sparse matrix. reference
- Dictionary of keys: (row, column)-pairs to the value of the elements.
- List of Lists: stores one list per row, with each entry containing the column index and the value.
- Coordinate List: a list of (row, column, value) tuples.
- Compressed Sparse Row: three (one-dimensional) arrays (A, IA, JA).
- Compressed Sparse Column: same as SCR

- example
- covert to sparse matrix
`python: csr_matrix(dense_matrix)`

- covert to dense matrix
`python: sparse_matrix.todense()`

- sparsity = 1.0 – count_nonzero(A) / A.size

- covert to sparse matrix

- Tensor
- multidimensional array.
- algriothm is similar to matrix
- dot product:
`python: tensordot()`

### Factorization

- Matrix Decompositions
- LU Decomposition
- square matrix
- , L is lower triangle matrix, U is upper triangle matrix. P matrix is used to permute the result or return result to the orignal order.
`python: lu(square_matrix)`

- QR Decomposition
- n*m matrix
- where Q a matrix with the size mm, and R is an upper triangle matrix with the size mn.
`python: qr(matrix)`

- Cholesky Decomposition
- square symmtric matrix where values are greater than zero
- , L is lower triangle matrix, U is upper triangle matrix.
- twice faster than LU decomposition.
`python: cholesky(matrix)`

- EigenDecomposition
- eigenvector: , is matrix we want to decomposite, is eigenvector, is eigenvalue(scalar)
- a matrix could have one eigenvector and eigenvalue for each dimension. So the matrix can be shown as prodcut of eigenvalues and eigenvectors. where Q is the matrix of eigenvectors, is the matrix of eigenvalue. This equotion also mean if we know eigenvalues and eigenvectors we can construct the orignal matrix.
`python: eig(matrix)`

- SVD(singluar value decomposition)
- , where A is m*n, U is m*m matrix, is m*m diagonal matrix also known as singluar value, is n*n matrix.
`python: svd(matrix)`

- reduce dimension
- select top largest singluar values in
- , where column select from , row selected from , B is approximate of the orignal matrix A.
- `python: TruncatedSVD(n_components=2)

- LU Decomposition

### Stats

- Multivari stats
- variance: ,
`python: var(vector, ddof=1)`

- standard deviation: ,
`python:std(M, ddof=1, axis=0)`

- covariance: ,
`python: cov(x,y)[0,1]`

- coralation: ,
**normorlized**to the value between -1 to 1.`python: corrcoef(x,y)[0,1]`

- PCA
- project high dimensions to subdimesnion
- steps:
- , which order by eigenvalue

- scikit learn

pca = PCA(2) # get two components

pca.fit(A)

print(pca.componnets_) # values

print(pca.explained_variance_) # vectors

B = pca.transform(A) # transform to new matrix

- Linear Regression

- , where b is coeffcient and unkown
- linear least squares( similar to MSE) , then . Issue: very slow
- MSE with SDG

- variance: ,

Reference: Basics of Linear Algebra for Machine Learning, **jason brownlee**, https://machinelearningmastery.com/linear_algebra_for_machine_learning/

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