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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\mathbb{R}^2, vector (3,2)(3, 2) expresses as a linear combination of standard basis vectors: 3(1,0)+2(0,1)3(1, 0) + 2(0, 1)
  • In polynomial space P2P_2, 2x23x+12x^2 - 3x + 1 expresses as a linear combination of basis {1,x,x2}\{1, x, x^2\}: 1(1)+(3)(x)+2(x2)1(1) + (-3)(x) + 2(x^2)
  • In matrix space M2x2M_{2x2}, (2113)\begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix} expresses as a linear combination of standard basis matrices
    • 2(1000)+1(0100)+(1)(0010)+3(0001)2\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + 1\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} + (-1)\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} + 3\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}

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\mathbb{R}^3 with standard basis {(1,0,0),(0,1,0),(0,0,1)}\{(1,0,0), (0,1,0), (0,0,1)\}, vector (2,3,1)(2,3,-1) has coordinate vector (231)\begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}
  • In P2P_2 with basis {1,1+x,1+x+x2}\{1, 1+x, 1+x+x^2\}, polynomial 2x+3x22-x+3x^2 has coordinate vector (233)\begin{pmatrix} 2 \\ -3 \\ 3 \end{pmatrix}
  • In R2\mathbb{R}^2 with basis {(1,1),(1,1)}\{(1,1), (1,-1)\}, vector (3,1)(3,1) has coordinate vector (21)\begin{pmatrix} 2 \\ 1 \end{pmatrix}

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=B1PAQ = B^{-1}PA, 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\mathbb{R}^2, changing from standard basis to basis {(1,1),(1,1)}\{(1,1), (1,-1)\} transforms vector (2,3)(2,3) to (2.50.5)\begin{pmatrix} 2.5 \\ -0.5 \end{pmatrix}
  • 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
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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