Math 2331 { Linear Algebra

The Dimension of a Vector SpaceMath 2331 Linear The Dimension of a Vector SpaceJiwen HeDepartment of Mathematics, University of jiwenhe/math2331 Jiwen He, University of HoustonMath 2331, Linear Algebra1 / The Dimension of a Vector SpaceDimension Basis The Dimension of a Vector SpaceThe Dimension of a Vector Space: TheoremsThe Dimension of a Vector Space: DefinitionThe Dimension of a Vector Space: ExampleDimensions of Subspaces ofR3 Dimensions of Subspaces: TheoremThe Basis TheoremDimensions of ColAand NulA: ExamplesJiwen He, University of HoustonMath 2331, Linear Algebra2 / The Dimension of a Vector SpaceDimension Basis TheoremThe Dimension of a Vector Space: TheoremsTheorem (9)If a vector spaceVhas a basis ={b1,..,bn}, then any set inVcontaining more thannvectors must be linearly :Suppose{u1.}

3 is a linear combination of v 1 and v 2, so by the Spanning Set Theorem, we may discard v 3. v 4 is not a linear combination of v 1 and v 2. So fv 1;v 2;v 4gis a basis for W. Also, dim W = . Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 14

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