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lecture_9 - MA 35100 LECTURE NOTES MONDAY FEBRUARY 1...

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Unformatted text preview: MA 35100 LECTURE NOTES: MONDAY, FEBRUARY 1 Geometric Interpretation of Matrix Multiplication Given a 2-dimensional vector ~x ∈ R 2 and a 2 × 2 matrix A , the product T ( ~x ) = A~x gives a new 2-dimensional vector ~ y = T ( ~x ). We discuss some examples to consider how linear transformations transform the plane R 2 . Denote the following three 2-dimensional vectors ~u = 1 , ~v = , and ~w = 2 . We will consider several types of transformations. To begin, consider the linear transformation T : R 2 → R 2 defined by T ( ~x ) =- 1 1 ~x that is T x 1 x 2 =- 1 1 x 1 x 2 =- x 2 x 1 . The three 2-dimensional vectors transform to T ( ~u ) = 1 , T ( ~v ) = , and T ( ~w ) =- 2 . Hence T is a rotation by 90 ◦ in the counterclockwise direction. With this example in mind, consider now the following six matrices: A = 2 0 0 2 B = 1 0 0 0 C =- 1 0 0 1 D = 0 1- 1 0 E = 1 0 . 5 1 F = 1- 1 1 1 Multiplication by A is a scaling by a factor of 2: A~u = 2 , A~v = , and A ~w = 4 . Multiplication by B is a projection onto the horizontal axis : B ~u = 1 , B~v = , and B ~w = ....
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lecture_9 - MA 35100 LECTURE NOTES MONDAY FEBRUARY 1...

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