Зміст курсу
Introduction to TensorFlow
2. Basics of TensorFlow
Introduction to TensorFlow
Basic Operations: Linear Algebra
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.
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.
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.
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:
Note
If you take two matrices with appropriate dimensions (
NxM @ MxK
, whereNxM
represents the dimensions of the first matrix andMxK
the second), and compute the dot product alongaxes=1
, it essentially performs matrix multiplication.
Завдання
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
andB
). - Using TensorFlow, find the inverse of matrix
A
. - Multiply the inverse of matrix
A
by matrixB
to find the solution matrixX
, which contains the values ofx
,y
, andz
.
Note
Slicing in TensorFlow operates similarly to NumPy. Therefore,
X[:, 0]
will retrieve all elements from the column at index0
. We will get to the slicing later in the course.
Все було зрозуміло?
Зміст курсу
Introduction to TensorFlow
2. Basics of TensorFlow
Introduction to TensorFlow
Basic Operations: Linear Algebra
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.
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.
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.
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:
Note
If you take two matrices with appropriate dimensions (
NxM @ MxK
, whereNxM
represents the dimensions of the first matrix andMxK
the second), and compute the dot product alongaxes=1
, it essentially performs matrix multiplication.
Завдання
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
andB
). - Using TensorFlow, find the inverse of matrix
A
. - Multiply the inverse of matrix
A
by matrixB
to find the solution matrixX
, which contains the values ofx
,y
, andz
.
Note
Slicing in TensorFlow operates similarly to NumPy. Therefore,
X[:, 0]
will retrieve all elements from the column at index0
. We will get to the slicing later in the course.
Все було зрозуміло?