C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes1-8

# C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes1-8 - 4...

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SECTION 1.8 ANOTHER WAY TO THINK ABOUT A x = b A x = b is a way to write a system of equations. Here’s the new way. Given a vector x , then A x is another vector y . When the matrix A is m × n , then we get a function or transformation T from R n , called the domain of T , to R m . For a vector x in R n , the vector T ( x ) = A x is the image of x . Finally, the set of all images T ( x ) is called the range of T . EXAMPLE. Suppose that A = 1 - 2 1 3 - 4 5 0 1 1 - 3 5 - 4 and b = 1 9 3 - 6 . Deﬁne the trans- formation T as above by T ( x ) = A x . Is b in the range of T ? If so, ﬁnd a vector x whose image under T is b and determine whether x is unique.

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The algebra of matrix-vector multiplication is fairly nice. In particular, A ( u + v ) = A u + A v and A ( c u ) = cA u . We can translate this nice algebra into properties of the transformation T : A transformation that has these properties is called linear. EXOTIC (FOR NOW) EXAMPLES. 1. Diﬀerentiation 2. Integration 3. Representing 3-D objects on a 2-D screen (computer graphics)
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Unformatted text preview: 4. Compressing photo images (gif, jpeg) 5. MP3 (I think) FACT. If T is a linear transformation, then 1. T ( ) = , and 2. T ( c 1 v 1 + c 2 v 2 + ··· + c p v p ) = c 1 T ( v 1 ) + c 2 T ( v 2 ) + ··· + c p T ( v p ) . In engineering Fact 2 is called the superposition principle. Example. Find all x in R 4 that are mapped into the zero vector by the linear transfor-mation x → A x for the matrix A = 1-2 1 0 3-4 5 2 1 1 1-3 5-4 1 . Example. When u and v are LI vectors in R n , then the plane P through u , v , and has parametric equation x = s u + t v . Show that a linear transformation T from R n to R m maps P onto a plane, or a line, or just the origin. HOMEWORK: SECTION 1.8...
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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes1-8 - 4...

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