2331_Notes_7o1_fill

2331_Notes_7o1_fill - Math 2331 Linear Algebra Section 7.1...

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Math 2331 Linear Algebra Section 7.1 Linear Transformations Part 1 Let V and W be vector spaces. A function : f VW is called a linear transformation if the following conditions hold 1. For all x and y in V, f(x+y) = f(x) + f(y) 2. For all x in V and all scalars k, f(kx) = k f(x) These conditions are called "preserving the operations" because Decide whether the following maps are linear transformations - Example 1: Let be defined by : f \\ ( ) 3 f xx = . Example 2: Let be defined by : f () 1 f = +
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Our normal concern in linear algebra are functions from for m and n greater than one. n \\ m 3 Example 3: Let be defined by 2 : f 2 3 x x f xy y y ⎡⎤ ⎛⎞ ⎢⎥ =− ⎜⎟ ⎣⎦ ⎝⎠ Example 4: Let be defined by 3 : T 2 x x Ty y z =
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Example 5: Let 44 12 21 3 : 3 0 T 3 2 x xx x T x ⎛⎞ ⎡⎤ ⎡ ⎜⎟ ⎢⎥ ⎢ + = ⎣⎦ ⎣ ⎝⎠ \\ Example 6: Let be defined by 2 : T 2 1 T yy + =
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Example 7: Let be defined by T(x) = Ax for 3 : T \\ 3 13 2 241 30 1 A ⎛⎞ ⎜⎟ = ⎝⎠
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2331_Notes_7o1_fill - Math 2331 Linear Algebra Section 7.1...

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