Linear Algebra - Introduction to Vector Spaces & Basis

Introduction to Vector Spaces & Basis

What is a Vector Space?

A vector space (also called a linear space) is a collection of vectors, which can be added together and multiplied (“scaled”) by numbers (called scalars in this context). Scalars are usually real numbers, but they can also be complex numbers or elements of other fields. Vector spaces are fundamental in linear algebra and are used to model various physical and abstract systems.

Formal Definition:

A set $V$ is called a vector space over a field $F$ (e.g., the field of real numbers $\mathbb{R}$ if it satisfies the following conditions:

1. Vector Addition: For any vectors $\mathbf{u}, \mathbf{v} \in V$, their sum $\mathbf{u} + \mathbf{v}$ is also in $V$.

2. Scalar Multiplication: For any vector $\mathbf{v} \in V$ and scalar $c \in F$, the product $c\mathbf{v}$ is also in $V$.

3. Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ for all $\mathbf{u}, \mathbf{v} \in V$.

4. Associativity of Addition: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$ for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$.

5. Associativity of Scalar Multiplication: $a(b\mathbf{v}) = (ab)\mathbf{v}$ for all $a, b \in F$ and $\mathbf{v} \in V$.

6. Distributivity of Scalar Multiplication:

   • $a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}$ for all $a \in F$ and $\mathbf{u}, \mathbf{v} \in V$.
   • $(a + b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}$ for all $a, b \in F$ and $\mathbf{v} \in V$.

7. Existence of Additive Identity: There exists a vector $\mathbf{0} \in V$ such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for all $\mathbf{v} \in V$.

8. Existence of Additive Inverses: For every $\mathbf{v} \in V$, there exists a vector $-\mathbf{v} \in V$ such that $\mathbf{v} + (-\mathbf{v}) = \mathbf{0}$.

Examples of Vector Spaces:

1. Euclidean Space: The set of all $n$-tuples of real numbers $\mathbb{R}^n$ is a vector space.
2. Polynomial Space: The set of all polynomials of a certain degree is a vector space.
3. Function Space: The set of all continuous functions on a given interval is a vector space.

What is a Subspace?

A subspace is a subset of a vector space that is also a vector space under the same addition and scalar multiplication as the original space. For a subset $W \subset V$ to be a subspace, it must satisfy three conditions:

1. Non-empty: $W$ contains the zero vector $\mathbf{0}$.
2. Closed under Addition: For any $\mathbf{u}, \mathbf{v} \in W$, the sum $\mathbf{u} + \mathbf{v}$ is also in $W$.
3. Closed under Scalar Multiplication: For any $\mathbf{v} \in W$ and scalar $c$, the product $c\mathbf{v}$ is in $W$.

Basis Vectors:

A basis of a vector space $V$ is a set of vectors that are linearly independent and span the entire vector space.

• Spanning: A set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ spans $V$ if every vector in $V$ can be written as a linear combination of these vectors.
• Linear Independence: The vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ are linearly independent if the only solution to the equation: [ c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_n\mathbf{v}_n = \mathbf{0} ] is $c_1 = c_2 = \dots = c_n = 0$.

If a vector space $V$ has a basis of $n$ vectors, then every vector in $V$ can be uniquely expressed as a linear combination of these basis vectors. The number $n$ is called the dimension of the vector space.

Significance of Basis:

1. Coordinate Systems: A basis allows for the representation of vectors in terms of coordinates relative to that basis.
2. Dimensionality: The number of vectors in the basis of a vector space determines its dimension.
3. Change of Basis: The concept of a basis is crucial when changing from one coordinate system to another, which is common in various applications like computer graphics and physics.

Examples of Basis:

1. Standard Basis in $\mathbb{R}^n$: The standard basis in $\mathbb{R}^n$ is the set of vectors $\{\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n\}$, where $\mathbf{e}_i$ is the vector with 1 in the $i$th position and 0 elsewhere.
2. Polynomial Basis: For the space of polynomials of degree at most 2, $\{1, x, x^2\}$ forms a basis.

Applications:

• Linear Transformations: Understanding vector spaces and basis vectors is essential for defining and analyzing linear transformations.
• Quantum Mechanics: In quantum mechanics, the state of a system is represented as a vector in a vector space, and the basis vectors represent possible states of the system.
• Differential Equations: Solutions to linear differential equations often form a vector space, and finding a basis for the solution space is a key task.