Lecturenotes25-28March2011

Lecturenotes25-28March2011 - Definition A matrix...

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Unformatted text preview: Definition: A matrix transformation T : R n → R m is said to be onto if evey vector in R m is the image of at least one vector in R n . Theorem 8.2.2: If T is a matrix transformation, T : R n-→ R n , then the following are equivalent (a) T is one-to-one (b) T is onto Example: Is the matrix transformation T : R 2 → R 3 , where T ( x,y ) = ( x,y,x + y ) is onto? Solution: T is onto if for any vector ( a,b,c ) ∈ R 3 we can find a corresponding ( x,y ) ∈ R 2 such that T ( x,y ) = ( a,b,c ) . From here we get linear system x = a y = b x + y = c T is onto if this sytem is consistent for all ( a,b,c ) . 1 0 a 0 1 b 1 1 c Row operations → 1 0 a 0 1 b 0 0 c- b- a So this system is consistent if c = a + b . Hence for (1 , 2 , 5) there is no ( x,y ) that is mapped to (1 , 2 , 5) under T. So T is not onto. Theorem 8.2.1 If T is a matrix transformation, T : R n-→ R m , then the following are equivalent (a) T is one-to-one (b) nullspace of [ T ] = { } Example: Consider the previous example we have [ T ] = 1 0 0 1 1 1 . Then null space of [ T ] is { } . Hence T is one-to-one....
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Lecturenotes25-28March2011 - Definition A matrix...

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