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MATA23H3 Lecture Notes - Lecture 10: Standard Basis, Empty Set, Maximal Independent Set

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School
UTSC
Department
Mathematics
Course
MATA23H3
Professor
Eric Moore

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=(cid:1871)(cid:1868)(cid:4666)[(cid:884), (cid:883),(cid:883),(cid:885)],[(cid:882),(cid:883),(cid:884), (cid:883)],[(cid:884),(cid:883),(cid:887),(cid:883)],[(cid:883),(cid:883),(cid:884), (cid:883)],[ (cid:883),(cid:885),(cid:885), (cid:887)](cid:4667) (cid:885) (cid:883) (cid:883) (cid:883) (cid:887)]~[(cid:883) (cid:883) (cid:883) (cid:883) (cid:884) (cid:1827)=[ (cid:1875) (cid:2869)(cid:1875) (cid:2870)(cid:1875) (cid:2871)(cid:1875) (cid:2872)(cid:1875) (cid:2873)]=[(cid:884) (cid:882) (cid:884) (cid:883) (cid:883) (cid:882) (cid:883) (cid:884) (cid:883) (cid:884) (cid:883) (cid:883) (cid:883) (cid:883) (cid:885) (cid:882) (cid:882) (cid:882) (cid:882) (cid:882)] (cid:882) (cid:882) (cid:882) (cid:883) (cid:883) (cid:884) (cid:883) (cid:887) (cid:884) (cid:885) ~[(cid:883) (cid:882) (cid:883) (cid:882) (cid:883) (cid:882) (cid:883) (cid:884) (cid:882) (cid:883) (cid:882)](cid:1870)(cid:1857)(cid:1856)(cid:1873)(cid:1855)(cid:1857)(cid:1856) (cid:1872)(cid:1867) (cid:1831)(cid:1832) (cid:882) (cid:882) (cid:882) (cid:883) (cid:883) (cid:882) (cid:882) (cid:882) (cid:882) (cid:1875) (cid:2871)=(cid:1875) (cid:2869)+(cid:884)(cid:1875) (cid:2870) (cid:1875) (cid:2873)=(cid:4666) (cid:883)(cid:4667)(cid:1875) (cid:2869)+(cid:1875) (cid:2870)+(cid:1875) (cid:2872) There is no pivot in column 3, 5 pivot in each column. Caution: the basis vectors come from the original matrix, not the echelon matrix. A set of vectors {(cid:1874)(cid:2869),(cid:1874)(cid:2870), ,(cid:1874)(cid:3038)} is independent iff the echelon form of (cid:1827)=[(cid:1874)(cid:2869),(cid:1874)(cid:2870), ,(cid:1874)(cid:3038)] has a. At most n independent vectors in (cid:3041) [(cid:883) (cid:882) (cid:883) (cid:883) (cid:882) (cid:883) (cid:882)] (cid:882) (cid:882) (cid:883) (cid:882) (cid:882) Put them as the column of a matrix and reduce to echelon form.

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Related Questions

please exaplin these clearly.

Show transcribed image text

For each of the following sets of vectors, decide whether the set is linearly dependent or

linearly independent. You should justify your answers completely. For those sets that are

linearly dependent, write one of the vectors as a linear combination of the others.

(a) {(1; 2; 3; 2); (2; 5; 5; 5); (2; 6; 4; 6)}.

(b) {(1; 1; 3; 4); (0; 2; 3; 1); (4; 0; 0; 2)}.

(c) {(0; 1; 1;-1); (k; 1; 1; 0); (1; 3; 3;-2)} (and how does the answer depend on k element of R?).

Don't we have to arrange the vectors in columns rather than rows? I asked this question before and expert answered with vectors in rows but theorem we have been taught says vectors must be arranged in columns and reduced to reo echelon form and then get rank to determin whether dependent or independent.