lect136_6_w09

lect136_6_w09 - Friday January 16 Lecture 6 Matrices as...

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Friday January 16 Lecture 6 : Matrices as linear transformations . (Refers to section 3.2) Expectations: 1. Define a transformation T from R n to R m . 2. Define the kernel and range of a linear transformation. 3. Given a matrix A , show that it can be viewed as a linear transformation T ( x ) = A x . 4. Define the standard matrix for a transformation T : R n R m . 5. If T ( x ) = A x , A standard matrix of T , then T maps R n onto R m 6.1 Definition A transformation is a function mapping R n to R m . A linear transformation T from R n to R m is a rule which assigns to each vector a in R n a unique vector denoted by T ( a ) in R m satisfying the condition: T ( α u + β v ) = α T ( u ) + β T ( v ) for any pair of vectors u and v in R n and scalars α and β . We sometimes say that T is linear if it “ respects linear combinations ”. 6.1.1 Definitions Given a linear transformation T : R n R m , R n is called the domain of T , and since T maps vectors into R m we call R m the codomain . T [ R n ] = { T ( u ) : u R n } is called the range of T or image of R n under T . The vector T ( u ) is the image of the vector u under T . 6.1.2 Example Verify whether the map f ( x ) = | x | is linear. What about the map g( x 1 , x 2 ) = ( x 1 2 , x 1 x 2 )? 6.1.3 Definition If T : R n R m is a transformation the set { u R n : T ( u ) = 0 } is called the kernel or the null space of T . 6.2 The matrix A m × n viewed as a linear transformation from R n into R m . Recall that from the properties of matrix algebra, given an matrix A m × n , any two vectors u and v in R n and any two scalars α and β we have
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A ( α u + β v ) = A ( α u ) + A ( β v ) = α A u + β A v a vector in R m . Hence the matrix A m × n can be viewed as a function sending a vector u in R n to a vector A u in R m . So the matrix A m × n can be viewed as function which " transforms " vectors in R n to vectors in R m . The matrix A m × n : R n R m is a transformation A : Not only that, it does so "linearly". It can be viewed as a linear transformation from R n into R m . When viewed in this way we call it a matrix transformation . 6.3 Examples 1) Consider the matrix A 1 0 0 1 Viewed as a linear transformation it will map vectors in R 2 to vectors in R 2 . We investigate to what vector it will map a vector ( x , y ). We see that 1 0 x x 0 1 y = y We see that this particular matrix "reflects" vectors in R 2 about the x -axis. This is why we call this particular matrix the "
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This note was uploaded on 09/04/2009 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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lect136_6_w09 - Friday January 16 Lecture 6 Matrices as...

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