Intuitive Way Definition A basis B for V is an independent generation set of V Is C a basis of V Independent yes Generation set difficult generates V Another way ID: 1001283
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1. Confirming that a set is a Basis
2. Intuitive WayDefinition: A basis B for V is an independent generation set of V.Is C a basis of V ?Independent?yesGeneration set?difficultgenerates V
3. Another wayGiven a subspace V, assume that we already know that dim V = k. Suppose S is a subset of V with k vectorsIf S is independentS is basisIf S is a generation setS is basisIs C a basis of V ?Independent?yesC is a basis of V Dim V = 2(parametric representation)C is a subset of V with 2 vectorsFind a basis for V
4. Another wayIf S is independentS is basisAssume that dim V = k. Suppose S is a subset of V with k vectorsBy the extension theorem, we can add more vector into S to form a basis.However, S already have k vectors, so it is already a basis.If S is a generation setS is basisBy the reduction theorem, we can remove some vector from S to form a basis.However, S already have k vectors, so it is already a basis.
5. ExampleIs B a basis of V ?Dim V = ? 3Independent set in V?yes B is a basis of V.
6. ExampleIs B a basis of V = Span S ? B is a subset of V with 3 vectors B is a basis of V.