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lecture_7

# lecture_7 - MA 35100 LECTURE NOTES WEDNESDAY JANUARY 27...

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MA 35100 LECTURE NOTES: WEDNESDAY, JANUARY 27 Linear Combinations (cont’d) We use the last fact from the previous lecture fact to prove a corollary which we will use repeatedly throughout the course: Theorem. Matrix multiplication preserves linear combinations. Explicitly, let A be an n × m matrix, and ~v = k 1 ~x 1 + k 2 ~x 2 + · · · + k r ~x r be a linear combination of vectors in R m . Then the vector A~v is a linear combination of vectors in R n : A~v = k 1 A ~x 1 + k 2 A ~x 2 + · · · + k r A ~x r . We explain why this is true. It suffices to show the following: Given two vectors ~x, ~ y R m and a scalar k R , we have a. A ( ~x + ~ y ) = A ~x + A ~ y b. A ( k ~x ) = k ( A ~x ). The reason why this suffices is we can express ~v = ~x + ~ y in terms of ~x = k r ~x r and ~ y = k 1 ~x 1 + k 2 ~x 2 + · · · + k r - 1 ~x r - 1 , then work inductively. Recall that we may write A~v = ~w 1 · ~v ~w 2 · ~v . . . ~w n · ~v where ~w j is the j th row of A . To show statement (a), we invoke the distributivity property of the dot product: A ( ~x + ~ y ) = ~w 1 · ( ~x + ~ y ) ~w 2 · ( ~x + ~ y ) . . . ~w n · ( ~x + ~ y ) = ~w 1 · ~x + ~w 1 · ~ y ~w 2 · ~x + ~w 2 · ~ y . . . ~w n · ~x + ~w n · ~ y = ~w 1 · ~x ~w 2 · ~x . . . ~w n · ~x + ~w 1 · ~ y ~w 2 · ~ y . . . ~w n · ~ y = A ~x + A ~ y. The proof of statement (b) is similar; we invoke the associativity property: A ( k ~x ) = ~w 1 · ( k ~x ) ~w 2 · ( k ~x ) . . . ~w n · ( k ~x ) = k ( ~w 1 · ~x ) k ( ~w 2 · ~x ) .

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lecture_7 - MA 35100 LECTURE NOTES WEDNESDAY JANUARY 27...

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