MATH 21 Lecture Notes - Lecture 6: Gaussian Elimination, Scalar Multiplication
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Subspace if: closed under addition, closed under scalar multiplication. Definition: sequence b is a basis if: span(b) = v (subspace of irn, b is linearly independent. Theorem: let w be a non-zero subspace of irn. Then w has a basis with at most n vectors. Example: determine if sequence of columns of matrix ((cid:883) (cid:884) (cid:883) (cid:884) (cid:887) (cid:884)) is a basis of ir3 (cid:884) (cid:885) (cid:885) Every column is a pivot column so the sequence of columns of the matrix is a basis of. Example: determine if (cid:4684)((cid:883)(cid:884) (cid:885)),((cid:884)(cid:885) (cid:887)),((cid:885)(cid:886) 7)(cid:4685) is a basis of ir3. ((cid:883) (cid:884) (cid:883) (cid:882) (cid:882) (cid:883)) (cid:882) (cid:883) (cid:882) ((cid:883) (cid:884) (cid:885) (cid:882) (cid:882) (cid:882)) (cid:882) (cid:883) (cid:884) The third column is not pivot column so the sequence is not a basis of ir3. Note: the third row of original vectors are all the negative of the sum of the two above numbers in ea(cid:272)h (cid:448)e(cid:272)tor.
