Vector spaces are the foundation of linear algebra. Bases and dimension help us understand their structure. A basis is a set of vectors that spans the space and is linearly independent. It's like a skeleton that defines the space's shape.

Dimension tells us how many vectors are in a basis. It's a key property of vector spaces, helping us compare and classify them. Understanding bases and dimension is crucial for solving linear systems and analyzing transformations between spaces.

Basis of a Vector Space

Definition and Key Properties

• A basis comprises a linearly independent subset of vectors that spans the entire vector space
• Multiple sets of vectors can form a basis for a given vector space
• Express every vector in the space as a unique linear combination of basis vectors
• Finite-dimensional vector spaces always have a finite number of basis vectors
• Removing any vector from a basis results in a set no longer spanning the space
• Basis provides a coordinate system allowing unique representation of vectors

Examples and Applications

• Standard basis for $\mathbb{R}^3$: $(1,0,0), (0,1,0), (0,0,1)$
• Polynomial basis for $P_2$: $\{1, x, x^2\}$
• Fourier basis for periodic functions: $\{1, \sin(x), \cos(x), \sin(2x), \cos(2x), ...\}$
• Basis for matrix space $M_{2x2}$: $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$

Basis Cardinality

Proof Concepts and Techniques

• Utilize linear independence and spanning properties of bases in the proof
• Apply the Replacement Theorem (Exchange Lemma) to transform one basis into another
• Maintain linear independence and spanning property during vector replacements
• Use contradiction to show different cardinalities violate basis definition
• Demonstrate invariance of basis vector count for a given vector space
• Establish foundation for vector space dimension concept

Proof Outline and Examples

• Start with two bases B₁ and B₂ of vector space V
• Assume |B₁| > |B₂| and derive a contradiction
  • Show a linear dependence in B₁ using vectors from B₂
  • Contradiction violates basis definition
• Repeat assuming |B₂| > |B₁| to show equality
• Example: Prove standard basis and diagonal matrix basis for $M_{2x2}$ have same cardinality
• Application: Prove dimension of $P_n$ (polynomials of degree ≤ n) is n+1

Finding a Basis

Methods and Techniques

• Apply Gram-Schmidt process to create orthogonal or orthonormal basis from linearly independent vectors
• Use Gaussian elimination for null space basis of linear equation systems
• Identify linearly independent columns for matrix column space basis
• Construct standard bases using monomials for polynomial vector spaces
• Eliminate linear dependencies among spanning vectors
• Employ Steinitz exchange lemma to extend linearly independent set or reduce spanning set

Examples and Applications

• Orthonormalize vectors $(1,1,0), (1,0,1), (0,1,1)$ in $\mathbb{R}^3$ using Gram-Schmidt
• Find basis for null space of matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{pmatrix}$
• Determine column space basis for matrix $B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 1 & 3 & 4 \end{pmatrix}$
• Construct basis for $P_3$ (polynomials of degree ≤ 3)
• Use Steinitz exchange to find basis of subspace spanned by $(1,1,1), (1,2,3), (2,3,4)$ in $\mathbb{R}^3$

Dimension of a Vector Space

Definition and Properties

• Dimension equals number of vectors in any basis of the space
• Finite-dimensional spaces have non-negative integer dimensions
• Zero vector space has dimension 0 (empty set basis)
• Subspace dimension ≤ parent vector space dimension
• Calculate dimension by finding a basis and counting its vectors
• Rank-nullity theorem relates vector space dimension to range and null space dimensions

Calculation Methods and Examples

• Determine dimension of $\mathbb{R}^n$ (n)
• Calculate dimension of $P_n$ (n+1)
• Find dimension of $M_{mxn}$ (m×n)
• Compute dimension of solution space for homogeneous system Ax = 0
• Use rank-nullity theorem to find nullity of linear transformation T: $\mathbb{R}^4$ → $\mathbb{R}^3$ with rank 2
• Calculate dimension of span{(1,1,0), (0,1,1), (1,0,1)} in $\mathbb{R}^3$