Linear Algebra Fundamentals: Comprehensive University-Level Notes on Vectors, Matrices, and Transformations

Linear Algebra Fundamentals
1. Vectors and Vector Spaces

1.1 Definition of Vectors

A vector is an element of a vector space, typically represented as a quantity with both
magnitude and direction. Vectors can be denoted as v⃗\vec{v}v or as an ordered tuple, such as
(v1,v2,...,vn)(v_1, v_2, ..., v_n)(v1​,v2​,...,vn​).

1.2 Properties of Vectors

Key vector properties:

● Addition: The sum of two vectors u⃗\vec{u}u and v⃗\vec{v}v is another vector
w⃗=u⃗+v⃗\vec{w} = \vec{u} + \vec{v}w=u+v.
● Scalar Multiplication: A vector can be scaled by a scalar aaa, producing a new vector
av⃗a\vec{v}av.
● Zero Vector: The zero vector 0⃗\vec{0}0 is a special vector with all components as zero,
(0,0,...,0)(0, 0, ..., 0)(0,0,...,0).
● Opposite Vector: For any vector v⃗\vec{v}v, there exists an opposite −v⃗-\vec{v}−v such
that v⃗+(−v⃗)=0⃗\vec{v} + (-\vec{v}) = \vec{0}v+(−v)=0.

1.3 Vector Operations

● Dot Product: For two vectors u⃗=(u1,u2,...,un)\vec{u} = (u_1, u_2, ...,
u_n)u=(u1​,u2​,...,un​) and v⃗=(v1,v2,...,vn)\vec{v} = (v_1, v_2, ..., v_n)v=(v1​,v2​,...,vn​), the
dot product is defined as u⃗⋅v⃗=u1v1+u2v2+...+unvn\vec{u} \cdot \vec{v} = u_1v_1 +
u_2v_2 + ... + u_nv_nu⋅v=u1​v1​+u2​v2​+...+un​vn​.
● Cross Product: In 3D, the cross product of vectors u⃗\vec{u}u and v⃗\vec{v}v produces a
vector orthogonal to both.

1.4 Vector Spaces

A vector space VVV over a field FFF is a set of vectors where vector addition and scalar
multiplication satisfy certain properties, such as closure, associativity, and distributivity.

2. Basis, Dimension, and Subspaces

2.1 Basis of a Vector Space

A basis of a vector space VVV is a set of linearly independent vectors that span VVV. Every
vector in VVV can be written uniquely as a linear combination of basis vectors.

, 2.2 Dimension of a Vector Space

The dimension of a vector space is the number of vectors in its basis. If VVV has a basis of nnn
vectors, then VVV is an nnn-dimensional vector space.

2.3 Subspaces

A subspace is a subset of a vector space that is itself a vector space under the same
operations. Important subspaces include:

● Span: The set of all linear combinations of a given set of vectors.
● Null Space: The set of all vectors that map to the zero vector under a given linear
transformation.

3. Matrices and Determinants

3.1 Definition of a Matrix

A matrix is a rectangular array of numbers arranged in rows and columns. It represents a linear
transformation and is used to solve systems of linear equations.

3.2 Matrix Operations

● Addition: Two matrices of the same dimension can be added element-wise.
● Scalar Multiplication: Every element of a matrix is multiplied by a scalar.
● Matrix Multiplication: For matrices AAA and BBB, the product ABABAB is computed by
taking the dot product of rows of AAA with columns of BBB.

3.3 Inverse of a Matrix

A matrix AAA has an inverse A−1A^{-1}A−1 if AA−1=A−1A=IAA^{-1} = A^{-1}A =
IAA−1=A−1A=I, where III is the identity matrix. Only square matrices with a non-zero
determinant have inverses.

3.4 Determinants

The determinant is a scalar value associated with square matrices, denoted as
det⁡(A)\det(A)det(A) or ∣A∣|A|∣A∣.

● Properties: Determinants are used to determine matrix invertibility (a matrix is invertible
if det⁡(A)≠0\det(A) \neq 0det(A)=0).
● Calculation: Determinants can be computed through cofactor expansion or row
reduction for larger matrices.

4. Eigenvalues and Eigenvectors