Linear Algebra Operations

TensorFlow offers a suite of functions dedicated to linear algebra operations, making matrix operations straightforward.

Matrix Multiplication

Here's a quick reminder of how matrix multiplication works.

There are two equivalent approaches for matrix multiplication:

The tf.matmul() function;
Using the @ operator.


              1234567891011121314
            
import tensorflow as tf

# Create two matrices
matrix1 = tf.constant([[1, 2], [3, 4], [2, 1]])
matrix2 = tf.constant([[2, 0, 2, 5], [2, 2, 1, 3]])

# Multiply the matrices
product1 = tf.matmul(matrix1, matrix2)
product2 = matrix1 @ matrix2

# Display tensors
print(product1)
print('-' * 50)
print(product2)

Note

Multiplying matrices of size 3x2 and 2x4 will give a matrix of 3x4.

Matrix Inversion

You can obtain the inverse of a matrix using the tf.linalg.inv() function. Additionally, let's verify a fundamental property of the inverse matrix.


              123456789101112131415
            
import tensorflow as tf

# Create 2x2 matrix
matrix = tf.constant([[1., 2.], [3., 4.]])

# Compute the inverse of a matrix
inverse_mat = tf.linalg.inv(matrix)

# Check the result
identity = matrix @ inverse_mat

# Display tensors
print(inverse_mat)
print('-' * 50)
print(identity)

Note

Multiplying a matrix with its inverse should yield an identity matrix, which has ones on its main diagonal and zeros everywhere else. Additionally, the tf.linalg module offers a wide range of linear algebra functions. For further details or more advanced operations, you might want to refer to its official documentation.

Transpose

You can obtain a transposed matrix using the tf.transpose() function.


              123456789101112
            
import tensorflow as tf

# Create a matrix 3x2
matrix = tf.constant([[1, 2], [3, 4], [2, 1]])

# Get the transpose of a matrix
transposed = tf.transpose(matrix)

# Display tensors
print(matrix)
print('-' * 40)
print(transposed)

Dot Product

You can obtain a dot product using the tf.tensordot() function. By setting up an axes argument you can choose along which axes to calculate a dot product. E.g. for two vectors by setting up axes=1 you will get the classic dot product between vectors. But when setting axes=0 you will get broadcasted matrix along 0 axes:


              1234567891011121314
            
import tensorflow as tf

# Create two vectors
matrix1 = tf.constant([1, 2, 3, 4])
matrix2 = tf.constant([2, 0, 2, 5])

# Compute the dot product of two tensors
dot_product_axes1 = tf.tensordot(matrix1, matrix2, axes=1)
dot_product_axes0 = tf.tensordot(matrix1, matrix2, axes=0)

# Display tensors
print(dot_product_axes1)
print('-' * 40)
print(dot_product_axes0)

Note

If you take two matrices with appropriate dimensions (NxM @ MxK, where NxM represents the dimensions of the first matrix and MxK the second), and compute the dot product along axes=1, it essentially performs matrix multiplication.

Task

Swipe to start coding

Background

A system of linear equations can be represented in matrix form using the equation:

AX = B

Where:

A is a matrix of coefficients.
X is a column matrix of variables.
B is a column matrix representing the values on the right side of the equations.

The solution to this system can be found using the formula:

X = A^-1 B

Where A^-1 is the inverse of matrix A.

Objective

Given a system of linear equations, use TensorFlow to solve it. You are given the following system of linear equations:

2x + 3y - z = 1.
4x + y + 2z = 2.
-x + 2y + 3z = 3.

Represent the system of equations in matrix form (separate it into matrices A and B).
Using TensorFlow, find the inverse of matrix A.
Multiply the inverse of matrix A by matrix B to find the solution matrix X, which contains the values of x, y, and z.

Note

Slicing in TensorFlow operates similarly to NumPy. Therefore, X[:, 0] will retrieve all elements from the column at index 0. We will get to the slicing later in the course.

Solution

Everything was clear?

Thanks for your feedback!

Section 1. Chapter 9

single

Ask AI

Ask anything or try one of the suggested questions to begin our chat

Swipe to show menu

Linear Algebra Operations

TensorFlow offers a suite of functions dedicated to linear algebra operations, making matrix operations straightforward.

Matrix Multiplication

Here's a quick reminder of how matrix multiplication works.

There are two equivalent approaches for matrix multiplication:

The tf.matmul() function;
Using the @ operator.


              1234567891011121314
            
import tensorflow as tf

# Create two matrices
matrix1 = tf.constant([[1, 2], [3, 4], [2, 1]])
matrix2 = tf.constant([[2, 0, 2, 5], [2, 2, 1, 3]])

# Multiply the matrices
product1 = tf.matmul(matrix1, matrix2)
product2 = matrix1 @ matrix2

# Display tensors
print(product1)
print('-' * 50)
print(product2)

Note

Multiplying matrices of size 3x2 and 2x4 will give a matrix of 3x4.

Matrix Inversion

You can obtain the inverse of a matrix using the tf.linalg.inv() function. Additionally, let's verify a fundamental property of the inverse matrix.


              123456789101112131415
            
import tensorflow as tf

# Create 2x2 matrix
matrix = tf.constant([[1., 2.], [3., 4.]])

# Compute the inverse of a matrix
inverse_mat = tf.linalg.inv(matrix)

# Check the result
identity = matrix @ inverse_mat

# Display tensors
print(inverse_mat)
print('-' * 50)
print(identity)

Note

Multiplying a matrix with its inverse should yield an identity matrix, which has ones on its main diagonal and zeros everywhere else. Additionally, the tf.linalg module offers a wide range of linear algebra functions. For further details or more advanced operations, you might want to refer to its official documentation.

Transpose

You can obtain a transposed matrix using the tf.transpose() function.


              123456789101112
            
import tensorflow as tf

# Create a matrix 3x2
matrix = tf.constant([[1, 2], [3, 4], [2, 1]])

# Get the transpose of a matrix
transposed = tf.transpose(matrix)

# Display tensors
print(matrix)
print('-' * 40)
print(transposed)

Dot Product


              1234567891011121314
            
import tensorflow as tf

# Create two vectors
matrix1 = tf.constant([1, 2, 3, 4])
matrix2 = tf.constant([2, 0, 2, 5])

# Compute the dot product of two tensors
dot_product_axes1 = tf.tensordot(matrix1, matrix2, axes=1)
dot_product_axes0 = tf.tensordot(matrix1, matrix2, axes=0)

# Display tensors
print(dot_product_axes1)
print('-' * 40)
print(dot_product_axes0)

Note

If you take two matrices with appropriate dimensions (NxM @ MxK, where NxM represents the dimensions of the first matrix and MxK the second), and compute the dot product along axes=1, it essentially performs matrix multiplication.

Task

Swipe to start coding

Background

A system of linear equations can be represented in matrix form using the equation:

AX = B

Where:

A is a matrix of coefficients.
X is a column matrix of variables.
B is a column matrix representing the values on the right side of the equations.

The solution to this system can be found using the formula:

X = A^-1 B

Where A^-1 is the inverse of matrix A.

Objective

Given a system of linear equations, use TensorFlow to solve it. You are given the following system of linear equations:

2x + 3y - z = 1.
4x + y + 2z = 2.
-x + 2y + 3z = 3.

Represent the system of equations in matrix form (separate it into matrices A and B).
Using TensorFlow, find the inverse of matrix A.
Multiply the inverse of matrix A by matrix B to find the solution matrix X, which contains the values of x, y, and z.

Note

Slicing in TensorFlow operates similarly to NumPy. Therefore, X[:, 0] will retrieve all elements from the column at index 0. We will get to the slicing later in the course.

Solution

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

Thanks for your feedback!

Section 1. Chapter 9

single

Basic Operations: Linear Algebra

Linear Algebra Operations

Matrix Multiplication

Matrix Inversion

Transpose

Dot Product

Background

Objective

Solution

Awesome!

Basic Operations: Linear Algebra

Linear Algebra Operations

Matrix Multiplication

Matrix Inversion

Transpose

Dot Product

Background

Objective

Solution

Awesome!