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Math 13

Linear algebra is a mathematical discipline focused on vector spaces and linear mappings, essential for various fields. It involves concepts such as vector spaces, linear transformations, and matrices, which have practical applications in areas like computer graphics, physics, and cryptography. Understanding these foundational elements is crucial for advanced studies in mathematics and its applications.

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14 views2 pages

Math 13

Linear algebra is a mathematical discipline focused on vector spaces and linear mappings, essential for various fields. It involves concepts such as vector spaces, linear transformations, and matrices, which have practical applications in areas like computer graphics, physics, and cryptography. Understanding these foundational elements is crucial for advanced studies in mathematics and its applications.

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An Overview of Linear Algebra

Your Name
February 26, 2025

1 Introduction
Linear algebra is the branch of mathematics concerning vector spaces and linear
mappings between these spaces. It is fundamental in many areas of mathematics
and its applications.

2 Vector Spaces
A vector space V over a field F consists of a set of elements called vectors,
along with operations of vector addition and scalar multiplication satisfying the
following properties:
• Associativity and commutativity of addition

• Existence of a zero vector and additive inverses


• Distributive and associative properties of scalar multiplication

3 Linear Transformations
A function T : V → W between vector spaces is a linear transformation if
for all u, v ∈ V and c ∈ F:

T (u + v) = T (u) + T (v), T (cv) = cT (v). (1)

4 Matrices
A matrix is a rectangular array of numbers that represents a linear transforma-
tion. Important operations include:
• Matrix addition and multiplication
• Determinants and inverses

• Eigenvalues and eigenvectors

1
5 Applications
Linear algebra has applications in:
• Computer graphics and machine learning
• Physics and engineering

• Cryptography and optimization

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