rev2023.3.1.43266. Let \(V\) be a subspace of \(\mathbb{R}^n\). This website is no longer maintained by Yu. Then \(A\) has rank \(r \leq n c__DisplayClass228_0.b__1]()", "4.02:_Vector_Algebra" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.03:_Geometric_Meaning_of_Vector_Addition" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.04:_Length_of_a_Vector" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.05:_Geometric_Meaning_of_Scalar_Multiplication" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.06:_Parametric_Lines" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.07:_The_Dot_Product" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.08:_Planes_in_R" : "property get [Map 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"09:_Vector_Spaces" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Some_Prerequisite_Topics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, 4.10: Spanning, Linear Independence and Basis in R, [ "article:topic", "license:ccby", "showtoc:no", "authorname:kkuttler", "licenseversion:40", "source@https://lyryx.com/first-course-linear-algebra" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FLinear_Algebra%2FA_First_Course_in_Linear_Algebra_(Kuttler)%2F04%253A_R%2F4.10%253A_Spanning_Linear_Independence_and_Basis_in_R, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Linearly Independent and Spanning Sets in \(\mathbb{R}^{n}\), Theorem \(\PageIndex{9}\): Finding a Basis from a Span, Definition \(\PageIndex{12}\): Image of \(A\), Theorem \(\PageIndex{14}\): Rank and Nullity, Definition \(\PageIndex{2}\): Span of a Set of Vectors, Example \(\PageIndex{1}\): Span of Vectors, Example \(\PageIndex{2}\): Vector in a Span, Example \(\PageIndex{3}\): Linearly Dependent Set of Vectors, Definition \(\PageIndex{3}\): Linearly Dependent Set of Vectors, Definition \(\PageIndex{4}\): Linearly Independent Set of Vectors, Example \(\PageIndex{4}\): Linearly Independent Vectors, Theorem \(\PageIndex{1}\): Linear Independence as a Linear Combination, Example \(\PageIndex{5}\): Linear Independence, Example \(\PageIndex{6}\): Linear Independence, Example \(\PageIndex{7}\): Related Sets of Vectors, Corollary \(\PageIndex{1}\): Linear Dependence in \(\mathbb{R}''\), Example \(\PageIndex{8}\): Linear Dependence, Theorem \(\PageIndex{2}\): Unique Linear Combination, Theorem \(\PageIndex{3}\): Invertible Matrices, Theorem \(\PageIndex{4}\): Subspace Test, Example \(\PageIndex{10}\): Subspace of \(\mathbb{R}^3\), Theorem \(\PageIndex{5}\): Subspaces are Spans, Corollary \(\PageIndex{2}\): Subspaces are Spans of Independent Vectors, Definition \(\PageIndex{6}\): Basis of a Subspace, Definition \(\PageIndex{7}\): Standard Basis of \(\mathbb{R}^n\), Theorem \(\PageIndex{6}\): Exchange Theorem, Theorem \(\PageIndex{7}\): Bases of \(\mathbb{R}^{n}\) are of the Same Size, Definition \(\PageIndex{8}\): Dimension of a Subspace, Corollary \(\PageIndex{3}\): Dimension of \(\mathbb{R}^n\), Example \(\PageIndex{11}\): Basis of Subspace, Corollary \(\PageIndex{4}\): Linearly Independent and Spanning Sets in \(\mathbb{R}^{n}\), Theorem \(\PageIndex{8}\): Existence of Basis, Example \(\PageIndex{12}\): Extending an Independent Set, Example \(\PageIndex{13}\): Subset of a Span, Theorem \(\PageIndex{10}\): Subset of a Subspace, Theorem \(\PageIndex{11}\): Extending a Basis, Example \(\PageIndex{14}\): Extending a Basis, Example \(\PageIndex{15}\): Extending a Basis, Row Space, Column Space, and Null Space of a Matrix, Definition \(\PageIndex{9}\): Row and Column Space, Lemma \(\PageIndex{1}\): Effect of Row Operations on Row Space, Lemma \(\PageIndex{2}\): Row Space of a reduced row-echelon form Matrix, Definition \(\PageIndex{10}\): Rank of a Matrix, Example \(\PageIndex{16}\): Rank, Column and Row Space, Example \(\PageIndex{17}\): Rank, Column and Row Space, Theorem \(\PageIndex{12}\): Rank Theorem, Corollary \(\PageIndex{5}\): Results of the Rank Theorem, Example \(\PageIndex{18}\): Rank of the Transpose, Definition \(\PageIndex{11}\): Null Space, or Kernel, of \(A\), Theorem \(\PageIndex{13}\): Basis of null(A), Example \(\PageIndex{20}\): Null Space of \(A\), Example \(\PageIndex{21}\): Null Space of \(A\), Example \(\PageIndex{22}\): Rank and Nullity, source@https://lyryx.com/first-course-linear-algebra, status page at https://status.libretexts.org. 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