5: Vector Space Rⁿ
- Page ID
- 58861
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: Subspaces and Spanning
- This page covers foundational linear algebra concepts, focusing on vector spaces, subspaces, and matrix transformations. It defines subspaces in \(\mathbb{R}^n\), including examples like the zero subspace and eigenspaces, and discusses spanning sets and the Span Theorem. Key concepts include the relationship between null space and image of matrices, illustrated through Gaussian elimination.
- 5.2: Independence and Dimension
- This page summarizes key concepts in linear algebra, focusing on linear independence and dependence of vectors in vector spaces, specifically \(\mathbb{R}^n\). It explains how to determine independence using combinations leading to zero, connects vector independence to matrix invertibility, and introduces concept of subspace dimensions and bases. The properties of subspaces are outlined, highlighting that dimensions can vary and relate to their containing spaces.
- 5.3: Orthogonality
- This page explores the extension of the dot product to \(\mathbb{R}^n\), defining vector length and orthogonality, along with properties such as commutativity and distributivity. It covers the Cauchy inequality, triangle inequality, and distance measures. Additionally, it highlights orthogonal and orthonormal sets, their linear independence, and the normalization process.
- 5.5: Similarity and Diagonalization
- This page covers the concepts of similar matrices and their diagonalization, defining similarity in terms of an invertible matrix \(P\). It emphasizes that diagonalizability relates to eigenvalues and eigenvectors, connecting distinct eigenvalues to linear independence. The page elaborates on eigenspaces, highlighting that a matrix is diagonalizable if the dimension of each eigenspace matches the eigenvalue's multiplicity.
- 5.6: Best Approximation and Least Squares
- This page presents various approaches to finding best approximations in linear algebra and applied mathematics. It covers linear approximations for systems of linear equations using normal equations, the least squares method for linear relationships, and polynomial approximations for data sets. Key concepts include minimizing squared errors, formulating and solving normal equations via Gaussian elimination, and ensuring unique solutions when matrices are invertible.
- 5.7: An Application to Correlation and Variance
- This page covers key statistical concepts, including sample data analysis, correlation, and sample statistics. It introduces sample mean, variance, and centered data, demonstrating their significance through vector representation and dot products. Correlation is explained using paired data, with caution against assuming causation from observed relationships. The text elaborates on correlation coefficients, their calculation, and the effects of data transformations on statistical measures.
- 5.E: Supplementary Exercises for Chapter 5
- This page covers vector spaces and linear independence in \(\mathbb{R}^n\), presenting statements that are verified as true or false with examples. It addresses properties of subspaces, conditions for linear independence, and characteristics of bases across dimensions. Specific cases involving vector sets and their relationships are explored to reinforce foundational linear algebra principles.