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extranotes Kernel and image

extranotes Kernel and image - Linear Algebra Notes Chapter...

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Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n × m matrix A = a 11 a 12 · · · a 1 m a 21 a 22 · · · a 2 m . . . . . . . . . a n 1 a n 2 · · · a nm and think of it as a function A : R m -→ R n . The kernel of A is defined as ker A = set of all x in R m such that A x = 0 . Note that ker A lives in R m . The image of A is im A = set of all vectors in R n which are A x for some x R m . Note that im A lives in R n . Many calculations in linear algebra boil down to the computation of kernels and images of matrices. Here are some different ways of thinking about ker A and im A . In terms of equations, ker A is the set of solution vectors x = ( x 1 , . . . , x m ) in R m of the n equations a 11 x 1 + a 12 x 2 + · · · + a 1 m x m = 0 a 21 x 1 + a 22 x 2 + · · · + a 2 m x m = 0 . . . a n 1 x 1 + a n 2 x 2 + · · · + a nm x m = 0 , (19a) and im A consists of those vectors y = ( y 1 , . . . y n ) in R n for which the system a 11 x 1 + a 12 x 2 + · · · + a 1 m x m = y 1 a 21 x 1 + a 22 x 2 + · · · + a 2 m x m = y 2 . . . a n 1 x 1 + a n 2 x 2 + · · · + a nm x m = y n , (19b) 1
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2 has a solution x = ( x 1 , . . . , x m ). A single equation a i 1 x 1 + a i 2 x 2 + · · · + a im x m = y i is called a hyperplane in R m . (So a line is a hyperplane in R 2 , and a plane is a hyperplane in R 3 .) Geometrically, ker A is the intersection of hyperplanes (19a), and im A is the set of vectors ( y 1 , . . . , y n ) R n for which the hyperplanes (19b) intersect in at least one point. If x and x are two solutions of (19b) for the same y , then A ( x - x ) = A x - A x = y - y = 0 , so x - x belongs to the kernel of A . If ker A = 0 (i.e., consists just of the zero vector) then there can be at most one solution. In general, the bigger the kernel, the more solutions there are to a given equation that has at least one solution. Thus, if x is one solution, then all other solutions are obtained from x by adding a vector from ker A .
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