Linear Algebra¶

What is Linear Algebra?¶

Linear algebra is the branch of mathematics concerning linear equations such as linear maps such as and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics See More

Applications of Linear Algebra in Data Science¶

Loss Functions
Regularization
Covariance Matrix
Support Vector Machine Classification
Principal Component Analysis (PCA)
Singular Value Decomposition
Word Embeddings
Latent Semantic Analysis (LSA)
Image Representation as Tensors
Convolution and Image Processing

SEE DETAILS: https://www.analyticsvidhya.com/blog/2019/07/10-applications-linear-algebra-data-science/

Linear Algebra Operations¶

# import numpy
import numpy as np 

Determinant of matrix¶

The determinant of a matrix is a special number that can be calculated from a square matrix \begin{equation} A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \ det(A) = ad - bc \end{equation} Whre, A is a $2 \times 2$ matrix.
np.linalg.det() - performs determinant of the matrix
Syntax: np.linalg.det(matrix)

Source: https://www.mathsisfun.com/algebra/matrix-determinant.html

# create matrix A
A = np.matrix("4, 5, 16, 7; 2,-3,2,3; 3,4,5,6; 4,7,8,9")
print(A)

[[ 4  5 16  7]
 [ 2 -3  2  3]
 [ 3  4  5  6]
 [ 4  7  8  9]]

# create matrix B
B = np.matrix("4,5,6,7;2,-3,3,3; 3,4,5,6; 4, 7,8,9")
print(B)

[[ 4  5  6  7]
 [ 2 -3  3  3]
 [ 3  4  5  6]
 [ 4  7  8  9]]

# determinant of A 
np.linalg.det(A) 

128.00000000000009

# determinant of B 
np.linalg.det(B)

Rank of matrix¶

np.linalg.matrix_rank() - returns rank of the matrix
Syntax: np.linalg.matrix_rank(matrix)

# rank of matrix A 
np.linalg.matrix_rank(A)

# rank of matrix B 
np.linalg.matrix_rank(B)

Inverse of a Matrix¶

np.linalg.inv() - returns the multiplicative inverse of a matrix.
Syntax: np.linalg.inv(matrix)

# inverse of matrix A 
np.linalg.inv(A)

matrix([[ 9.37500000e-02, -4.68750000e-01,  3.68750000e+00,
         -2.37500000e+00],
        [ 3.53252781e-17, -2.50000000e-01,  5.00000000e-01,
         -2.50000000e-01],
        [ 9.37500000e-02,  3.12500000e-02, -3.12500000e-01,
          1.25000000e-01],
        [-1.25000000e-01,  3.75000000e-01, -1.75000000e+00,
          1.25000000e+00]])

# inverse of matrix B 
np.linalg.inv(B)

System of linear equations¶

Consider a system of equations $$3x + y + 2z = 2$$3x + 2y + 5z = -1$$6x + 7y + 8z = 3$$

Now we can write the equations in the form of $Ax = b$

np.linalg.matrix_rank() - returns rank of the matrix
Syntax: np.linalg.matrix_rank(matrix)

### Rank of matrix 
- `np.linalg.matrix_rank()` - returns rank of the matrix 
- **Syntax:** `np.linalg.matrix_rank(matrix)`

Introduction to NumPy