Introductions to Linear Transformations Function T that maps a vector space V into a vector space W: V: the domain of T W: the codomain of T Chapter 6 Linear Transformations

6-2 Image of v under T: If v is in V and w is in W such that Then w is called the image of v under T. the range of T: The set of all images of vectors in V. the preimage of w: The set of all v in V such that T(v)=w.

6-3 (1) A linear transformation is said to be operation preserving, because the same result occurs whether the operations of addition and scalar multiplication are performed before or after T. Addition in V Addition in W Scalar multiplication in V Scalar multiplication in W (2) A linear transformation from a vector space into itself is called a linear operator. Notes:

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6-5 Two simple linear transformations: Zero transformation: Identity transformation:

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The Kernel and Range a Linear Transformation

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6-10 Note: The kernel of T is sometimes called the nullspace of T.

6-11 Range of a linear transformation T:

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6-15 One-to-one: one-to-onenot one-to-one Onto: (T is onto W when W is equal to the range of T.)

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Matrices for Liner Transformations Three reasons for matrix representation of a linear transformation: – It is simpler to write. – It is simpler to read. – It is more easily adapted for computer use. Two representations of the linear transformation T:R 3 →R 3 :

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6-21 Notes: (1) The standard matrix for the zero transformation from R n into R m is the m  n zero matrix. (2) The standard matrix for the identity transformation from R n into R n is the n  n identity matrix I n Composition of T 1 :R n →R m with T 2 :R m →R p :

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6-23 Note: If the transformation T is invertible, then the inverse is unique and denoted by T –1.

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Transition Matrices and Similarty

6-27 Two ways to get from to :

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6-29 Note: From the definition of similarity it follows that any tow matrices that represent the same linear transformation with respect to different based must be similar.

Applications of Linear Transformations

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