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
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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 R 3 \mathbb{R}^3 R 3 : ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) (1,0,0), (0,1,0), (0,0,1) ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 )
Polynomial basis for P 2 P_2 P 2 : { 1 , x , x 2 } \{1, x, x^2\} { 1 , x , x 2 }
Fourier basis for periodic functions: { 1 , sin ( x ) , cos ( x ) , sin ( 2 x ) , cos ( 2 x ) , . . . } \{1, \sin(x), \cos(x), \sin(2x), \cos(2x), ...\} { 1 , sin ( x ) , cos ( x ) , sin ( 2 x ) , cos ( 2 x ) , ... }
Basis for matrix space M 2 x 2 M_{2x2} M 2 x 2 : ( 1 0 0 0 ) , ( 0 1 0 0 ) , ( 0 0 1 0 ) , ( 0 0 0 1 ) \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} ( 1 0 0 0 ) , ( 0 0 1 0 ) , ( 0 1 0 0 ) , ( 0 0 0 1 )
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 2 x 2 M_{2x2} M 2 x 2 have same cardinality
Application: Prove dimension of P n P_n 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 ) (1,1,0), (1,0,1), (0,1,1) ( 1 , 1 , 0 ) , ( 1 , 0 , 1 ) , ( 0 , 1 , 1 ) in R 3 \mathbb{R}^3 R 3 using Gram-Schmidt
Find basis for null space of matrix A = ( 1 2 3 2 4 6 ) A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{pmatrix} A = ( 1 2 2 4 3 6 )
Determine column space basis for matrix B = ( 1 2 3 0 1 1 1 3 4 ) B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 1 & 3 & 4 \end{pmatrix} B = 1 0 1 2 1 3 3 1 4
Construct basis for P 3 P_3 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 ) (1,1,1), (1,2,3), (2,3,4) ( 1 , 1 , 1 ) , ( 1 , 2 , 3 ) , ( 2 , 3 , 4 ) in R 3 \mathbb{R}^3 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 R n \mathbb{R}^n R n (n)
Calculate dimension of P n P_n P n (n+1)
Find dimension of M m x n M_{mxn} M m x n (m×n)
Compute dimension of solution space for homogeneous system Ax = 0
Use rank-nullity theorem to find nullity of linear transformation T: R 4 \mathbb{R}^4 R 4 → R 3 \mathbb{R}^3 R 3 with rank 2
Calculate dimension of span{(1,1,0), (0,1,1), (1,0,1)} in R 3 \mathbb{R}^3 R 3