MATH115 Lecture Notes - Lecture 32: Horse Length, Triangular Matrix, Scalar Multiplication
10 Oct 2015
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Tuesday, november 18 lecture 33: subspaces and spanning families. [0,1] is a subspace of the vector space of all functions on [0,1]: define the trivial subspace, prove that the intersection of two subspaces is a subspace. 33. 1 definition we say that a subset w of a vector space v is a subspace of v if: w contains its zero-vector and, w is closed under addition and scalar multiplication (equivalently is closed under linear combinations). Yes, since, along with the two properties 1 and 2 above, the subspace w inherits all the addition and scalar multiplication properties from the vector space v that contains it. So it is important to remember : any non-empty subset w of the vector space (v, +, ) which satisfies the 2 closure axioms is itself a vector space. There is no need to check that (w, +, ) satisfies the other axioms (they automatically do).
