c2s3 - 2.3 Linear Transformations of Euclidean Spaces 1...

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Unformatted text preview: 2.3 Linear Transformations of Euclidean Spaces 1 Chapter 2. Dimension, Rank, and Linear Transformations 2.3 Linear Transformations of Euclidean Spaces Definition. A linear transformation T : R n R m is a function whose domain is R n and whose codomain is R m , where (1) T ( ~u + ~v ) = T ( ~u ) + T ( ~v ) for all ~u,~v R n , and (2) T ( r~u ) = rT ( ~u ) for all ~u R n and for all r R . Note. Combining (1) and (2) gives T ( r~u + s~v ) = rT ( ~u ) + sT ( ~v ) for all ~u,~v R n and r, s R . As the book says, linear transformations preserve linear combinations. Note. T ( ~ 0) = T (0 ~ 0) = 0 T ( ~ 0) = ~ . Example. Page 152 number 4. 2.3 Linear Transformations of Euclidean Spaces 2 Example. Page 145 Example 4. Notice that every linear transformation of R R is of the form T ( x ) = ax. The graphs of such functions are lines through the origin. Theorem 2.7. Bases and Linear Transformations....
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c2s3 - 2.3 Linear Transformations of Euclidean Spaces 1...

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