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9: Change of Basis

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    58889

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    If \(A\) is an \(m \times n\) matrix, the corresponding matrix transformation \(T_A : \mathbb{R}^n \to \mathbb{R}^m\) is defined by

    \[T_A (\mathbf{x}) = A \mathbf{x} \quad \mbox{ for all columns } \mathbf{x} \mbox{ in } \mathbb{R}^n \nonumber \]

    It was shown in Theorem [thm:005789] that every linear transformation \(T : \mathbb{R}^n \to \mathbb{R}^m\) is a matrix transformation; that is, \(T = T_A\) for some \(m \times n\) matrix \(A\). Furthermore, the matrix \(A\) is uniquely determined by \(T\). In fact, \(A\) is given in terms of its columns by

    \[A = \left[ \begin{array}{cccc} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \cdots & T(\mathbf{e}_n) \end{array}\right] \nonumber \]

    where \(\{\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n\}\) is the standard basis of \(\mathbb{R}^n\).

    In this chapter we show how to associate a matrix with any linear transformation \(T : V \to W\) where \(V\) and \(W\) are finite-dimensional vector spaces, and we describe how the matrix can be used to compute \(T(\mathbf{v})\) for any \(\mathbf{v}\) in \(V\). The matrix depends on the choice of a basis \(B\) in \(V\) and a basis \(D\) in \(W\), and is denoted \(M_{DB}(T)\). The case when \(W = V\) is particularly important. If \(B\) and \(D\) are two bases of \(V\), we show that the matrices \(M_{BB}(T)\) and \(M_{DD}(T)\) are similar, that is \(M_{DD}(T) = P^{-1}M_{BB}(T)P\) for some invertible matrix \(P\). Moreover, we give an explicit method for constructing \(P\) depending only on the bases \(B\) and \(D\). This leads to some of the most important theorems in linear algebra, as we shall see in Chapter [chap:11].

    • 9.1: The Matrix of a Linear Transformation
      This page provides an overview of representing linear transformations as matrix multiplications involving specific bases for vector spaces. It defines the matrix \(M_{DB}(T)\) for transformations between bases and highlights the crucial role of coordinate vectors. The relationship between matrix operations and linear transformations is explored, alongside properties such as composition and isomorphism conditions linked to matrix invertibility.
    • 9.2: Operators and Similarity
      This page covers linear transformations and their matrix representations in vector spaces, emphasizing the role of bases in simplifying calculations. Key concepts include the definitions of linear operators, the similarity of matrices across different bases, diagonalization, and the computation of characteristic polynomials.
    • 9.3: Invariant Subspaces and Direct Sums
      This page covers \(T\)-invariant subspaces in linear algebra, emphasizing their importance in representing linear operators and matrix forms, particularly block upper triangular matrices. It explains relationships between eigenvalues, eigenvectors, and matrix representations, alongside the concepts of direct sums and dimensions of vector spaces.

    This page titled 9: Change of Basis is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by W. Keith Nicholson (Lyryx Learning Inc.) via source content that was edited to the style and standards of the LibreTexts platform.