Linear Independence. Linear Transformations. Matrix Operations

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Astana IT University
Department of Computational and Data Science
Linear Independence
Linear Transformations
Matrix Operations
Astana, 2024
Linear Algebra
1

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Astana IT University
Week 2
Linear Independence
Linear Transformations
The Matrix of a Linear Transformation
Matrix Operations
Examples and Exercises
Astana, 2024
Linear Algebra
2

3.

Astana IT University
Linear Independence
The homogeneous equations can be studied from a different
perspective by writing them as vector equations. In this way, the
focus shifts from the unknown solutions of Ax=0 to the vectors
that appear in the vector equations.
Astana, 2024
Linear Algebra
3

4.

Astana IT University
Linear Independence
Astana, 2024
Linear Algebra
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Astana IT University
Linear Independence
Astana, 2024
Linear Algebra
5

6.

Astana IT University
Linear Independence
Astana, 2024
Linear Algebra
6

7.

Astana IT University
Linear Transformations
Astana, 2024
Linear Algebra
7

8.

Astana IT University
Linear Transformations
Astana, 2024
Linear Algebra
8

9.

Astana IT University
The Matrix of a Linear Transformation
Astana, 2024
Linear Algebra
9

10.

Astana IT University
The Matrix of a Linear Transformation
Astana, 2024
Linear Algebra
10

11.

Astana IT University
The Matrix of a Linear Transformation
Astana, 2024
Linear Algebra
11

12.

Astana IT University
The Matrix of a Linear Transformation
Astana, 2024
Linear Algebra
12

13.

Astana IT University
The Matrix of a Linear Transformation
Astana, 2024
Linear Algebra
13

14.

Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
14

15.

Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
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16.

Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
16

17.

Astana IT University
Matrix Operations
Multiplication of matrices corresponds to composition of
linear transformations.
Astana, 2024
Linear Algebra
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Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
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Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
19

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Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
20

21.

Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
21

22.

Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
22

23.

Astana IT University
Matrix Operations
Astana, 2024
Linear Algebra
23

24.

Astana IT University
Exercises
1) Determine if the columns of the matrix form a linearly
independent set
2) Compute A+B, AB, BA
Astana, 2024
Linear Algebra
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