MATH-M 303 Lecture 19: M303 4.5 Notes (Nov. 16)
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M303 section 4. 5 notes- the dimension of a vector space. Theorem 9: proof, proof: row would lack pivot linearly independent and span, note: for nonzero vector space, infinite number of bases exist (scalar multiples of basis are still. In notation, want to determine if #(cid:1828)(cid:2869)=#(cid:1828)(cid:2870: for = , this is true; all bases must have (cid:1866) elements (matrix must have (cid:1866) vectors in ) to be. Theorem 9- if is a vector space and (cid:1828)={(cid:2778),(cid:2779), ,} is a basis with (cid:1866) elements, then any set in with more than (cid:1866) elements must be linearly dependent. (cid:1866) (cid:1868) matrix formed with these vectors would have more columns than rows, so at least one. Coordinate vectors linearly dependent; using invertible map that sends (cid:2203) to coordinate vectors, can conclude based on dependence of coordinate vectors that (cid:2203) linearly dependent as well. Theorem 10- if vector space has (cid:1828)(cid:2869),(cid:1828)(cid:2870) are bases, then #(cid:1828)(cid:2869)=#(cid:1828)(cid:2870)
