Coordinate vectors and change of basis are key concepts in linear algebra. They allow us to represent vectors in different ways and switch between different coordinate systems. This flexibility is crucial for solving problems and understanding vector spaces.
These ideas build on earlier topics like linear combinations and bases. They show how the same vector can be described differently depending on the chosen basis, connecting abstract vector spaces to concrete representations.
Vectors as Linear Combinations
Linear Combinations and Basis Vectors
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Linear combination involves summing scalar multiples of vectors
Basis of a vector space constitutes a linearly independent set spanning the entire space
Any vector in the space uniquely expresses as a linear combination of basis vectors
Coefficients in the linear combination form coordinates of the vector relative to the given basis
Finding coefficients requires solving a system of linear equations
Number of basis vectors equals the dimension of the vector space
Different bases for the same vector result in varying coordinate representations
Examples of Linear Combinations
In R2, vector (3,2) expresses as a linear combination of standard basis vectors: 3(1,0)+2(0,1)
In polynomial space P2, 2x2−3x+1 expresses as a linear combination of basis {1,x,x2}: 1(1)+(−3)(x)+2(x2)
In matrix space M2x2, (2−113) expresses as a linear combination of standard basis matrices
2(1000)+1(0010)+(−1)(0100)+3(0001)
Coordinate Vectors of Basis
Determining Coordinate Vectors
Coordinate vector contains coefficients of linear combination expressing a vector in terms of given basis
Set up equation equating given vector to linear combination of basis vectors with unknown coefficients
Convert equation into system of linear equations and solve for unknown coefficients
Resulting coefficients form entries of coordinate vector
Dimension of coordinate vector equals number of basis vectors
Coordinate vector remains unique for a given vector and basis
Zero vector always has coordinate vector of all zeros, regardless of chosen basis
Examples of Coordinate Vectors
In R3 with standard basis {(1,0,0),(0,1,0),(0,0,1)}, vector (2,3,−1) has coordinate vector 23−1
In P2 with basis {1,1+x,1+x+x2}, polynomial 2−x+3x2 has coordinate vector 2−33
In R2 with basis {(1,1),(1,−1)}, vector (3,1) has coordinate vector (21)
Transition Matrices for Bases
Constructing Transition Matrices
Transition matrix transforms coordinates of vector from one basis to another
Express each vector of basis B as linear combination of vectors in basis A to find transition matrix from A to B
Coefficients of linear combinations form columns of transition matrix
Transition matrix from A to B equals inverse of transition matrix from B to A
Dimension of transition matrix equals dimension of vector space
Determinant of transition matrix remains non-zero, representing change between bases
Transition matrices compose to represent changes between multiple bases
Properties and Applications of Transition Matrices
Transition matrices always remain invertible
Columns of transition matrix represent coordinate vectors of new basis in terms of old basis
Transition matrices preserve linear independence and dependence of sets of vectors
Composition of transition matrices follows matrix multiplication rules
Transition matrices apply in various fields (computer graphics, physics, engineering)
Transition matrices facilitate basis-dependent calculations in linear algebra
Change of Basis Transformations
Transforming Coordinate Vectors and Matrices
Change coordinate vector from basis A to B by multiplying transition matrix from A to B by coordinate vector with respect to A
Transform matrix representation P of linear transformation T from basis A to B using formula Q=B−1PA, where B represents transition matrix from A to B
Changing basis simplifies computations by transforming matrices into more convenient forms (diagonal matrices)
Trace and determinant of linear transformation's matrix representation remain invariant under change of basis
Eigenvalues of matrix preserve under change of basis, while eigenvectors transform according to change of basis
Change of basis finds canonical forms of matrices (Jordan canonical form)
Applications include coordinate transformations in physics or computer graphics
Examples of Change of Basis
In R2, changing from standard basis to basis {(1,1),(1,−1)} transforms vector (2,3) to (2.5−0.5)
Diagonalization process involves change of basis to eigenvector basis, simplifying matrix computations
Fourier transform represents change of basis from time domain to frequency domain in signal processing
In quantum mechanics, changing from position basis to momentum basis transforms wave functions