Glossary

1.6 Coordinate vectors and change of basis

4 min read•august 16, 2024

Coordinate vectors and change of 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 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 vectors: 3(1,0)+2(0,1)3(1, 0) + 2(0, 1)
  • In polynomial space P2P_2, 2x2−3x+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}, (21−13)\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

  • 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 (23−1)\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 2−x+3x22-x+3x^2 has coordinate vector (2−33)\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

  • 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 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 T from basis A to B using formula Q=B−1PAQ = 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 (2.5−0.5)\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

Key Terms to Review (18)

[v]_b: [v]_b is the coordinate vector of a vector v with respect to a basis b. This term represents how the vector can be expressed as a linear combination of the basis vectors, allowing us to transition between different bases in vector spaces. Understanding coordinate vectors is crucial for performing operations such as linear transformations and changing bases efficiently.
[x]_{c}: [x]_{c} is a notation that represents the coordinate vector of a vector x with respect to a basis c in a vector space. This means it expresses the vector x as a linear combination of the basis vectors in c, allowing us to easily understand and manipulate the vector in different coordinate systems. This notation is crucial for working with change of basis, as it helps us transition between various representations of vectors in linear algebra.
Basis: A basis is a set of vectors in a vector space that are linearly independent and span the entire space. This means that any vector in the space can be expressed as a linear combination of the basis vectors. Understanding the concept of a basis is crucial because it helps define the structure of a vector space, connecting ideas like linear independence, dimension, and coordinate systems.
Change of Basis Matrix: A change of basis matrix is a square matrix that transforms the coordinates of a vector from one basis to another in a vector space. This concept is vital because it allows for the representation of vectors in different coordinate systems, making it easier to understand their properties and relationships in various contexts. The ability to switch between bases is essential for simplifying computations and revealing the geometric interpretation of linear transformations.
Change of Basis Theorem: The change of basis theorem provides a method to convert the coordinate representation of a vector from one basis to another in a vector space. It highlights how different bases can describe the same vector in different ways, allowing for more flexibility in calculations and interpretations within linear algebra.
Coordinate transformation: A coordinate transformation is a mathematical operation that changes the representation of a vector from one coordinate system to another. This process is crucial for understanding how vectors relate to different bases, as it allows us to express the same geometric object in various ways depending on the chosen framework. By converting coordinates, we can simplify calculations, understand geometric relationships, and explore linear mappings more effectively.
Coordinate Vector: A coordinate vector is a representation of a vector in relation to a specific basis of a vector space. It expresses the vector as a linear combination of the basis vectors, capturing the components needed to reconstruct the vector using those basis elements. Understanding coordinate vectors is essential for transitioning between different bases and facilitates operations like linear transformations and changes of basis.
Dimensionality Theorem: The Dimensionality Theorem states that in a finite-dimensional vector space, the dimensions of the null space and the range of a linear transformation add up to equal the dimension of the domain of that transformation. This theorem highlights the relationship between the rank and nullity of a linear transformation, providing a crucial understanding of how different vector spaces are interconnected.
Image representation: Image representation refers to how a vector is expressed in terms of a specific basis, particularly when considering linear transformations. It is crucial for understanding how different bases can change the way vectors are perceived and calculated, making it essential for operations like changing bases and finding coordinate vectors in linear algebra.
Linear transformation: A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means if you take any two vectors and apply the transformation, the result will be the same as transforming each vector first and then adding them together. It connects to various concepts, showing how different bases interact, how they can change with respect to matrices, and how they impact the underlying structure of vector spaces.
Linearly independent: Linearly independent refers to a set of vectors in a vector space that cannot be expressed as a linear combination of each other. This means that none of the vectors in the set can be represented as a combination of the others, which is crucial for defining unique coordinate representations and understanding changes of basis.
Matrix Multiplication: Matrix multiplication is a binary operation that produces a matrix from two matrices by multiplying the rows of the first matrix by the columns of the second matrix. This operation is fundamental in linear algebra and connects directly to various important concepts like coordinate transformations, the behavior of linear transformations, and dimensionality reduction in data analysis.
Orthonormal Basis: An orthonormal basis is a set of vectors in a vector space that are both orthogonal to each other and normalized to have a length of one. This concept is fundamental in understanding the structure of vector spaces and facilitates easier calculations, especially when dealing with projections, transformations, and inner product spaces.
Scalar Multiplication: Scalar multiplication is an operation that involves multiplying a vector by a scalar (a real number), which results in a new vector that points in the same or opposite direction depending on the sign of the scalar and has its magnitude scaled accordingly. This operation is fundamental to understanding how vectors behave within vector spaces, as it helps define their structure and properties. It also plays a crucial role when discussing coordinate vectors and changes of basis, as it allows for the transformation and manipulation of vectors within different coordinate systems.
Span: Span refers to the set of all possible linear combinations of a given set of vectors. It represents all the points that can be reached in a vector space through these combinations, effectively capturing the extent of coverage these vectors have within that space. The concept of span connects deeply with understanding vector spaces, the relationships between vectors regarding independence and dependence, how coordinates shift during basis changes, and the creation of orthogonal sets in processes like Gram-Schmidt.
Standard Basis: The standard basis is a specific set of vectors that provides a reference for all other vectors in a given vector space. In $ ext{R}^n$, the standard basis consists of the unit vectors $ ext{e}_1, ext{e}_2, ..., ext{e}_n$, where each vector has a 1 in one coordinate and 0s in all others. This basis is crucial for understanding how vectors can be expressed in terms of coordinates and how transformations between different bases can occur.
Transition Matrix: A transition matrix is a matrix that describes the transformation of coordinate vectors when changing from one basis to another in a vector space. It provides the necessary information to convert vectors represented in one basis to their corresponding representations in another basis, making it an essential tool for understanding how different coordinate systems relate to each other.
Vector Addition: Vector addition is the process of combining two or more vectors to produce a new vector. This operation is fundamental in vector spaces, as it allows for the exploration of properties like closure and linear combinations. Understanding vector addition also lays the groundwork for working with coordinate vectors, where it helps in visualizing and manipulating vectors in different bases.
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