la_m7_handouts

# la_m7_handouts - MAC 2103 Module 7 Euclidean Vector Spaces...

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1 1 MAC 2103 Module 7 Euclidean Vector Spaces II 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Determine if a linear operator in n is one-to-one. 2. Find the inverse of a linear operator in n . 3. Use the images of the standard basis vectors to find a standard matrix in n . 4. Find the polynomial q=T(p) in P 1 corresponding to the transformation T on any polynomials in P 1 . http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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2 3 Rev.09 Euclidean Vector Spaces II http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Properties of Linear Properties of Linear Transformations from Transformations from n to to m Linear Transformations and Polynomials Linear Transformations and Polynomials There are two major topics in this module: 4 Rev.F09 What are the Important Properties of Linear Transformations? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. A transformation T: n m is linear if both of the following relationships hold for all vectors u and v in n and for every scalar s : (See Theorem 4.3.2) a) T( u + v ) = T( u ) + T( v ) b) T( s u ) = s T( u)
3 5 Rev.F09 What are the Important Properties of Linear Transformations? (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. It follows that: T(- v ) = T[(-1) v ] = (-1)T( v ) = - T(v), T( u - v ) = T[ u + (-1) v ] = T( u ) + (-1)T( v ) = T( u ) - T( v ), T( 0 ) = T(0 v ) = 0T( v ) = 0 , since 0 v = 0 ; and T ( ! v ) = T ( s 1 ! v 1 + s 2 ! v 2 + ... + s n ! v n ) = s 1 T ( ! v 1 ) + s 2 T ( ! v 2 ) + ... + s n T ( ! v n ), ! v = s 1 ! v 1 + s 2 ! v 2 + ... + s n ! v n 6 Rev.F09 How to Determine if a Linear Operator in n is one-to-one? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Example: Find the standard matrix for the linear operator defined by the equations and determine whether the operator is one-to-one? (a) Solution: w 1 = 6 x 1 ! 3 x 2 w 2 = 2 x 1 ! x 2 w 1 w 2 ! " # # \$ % = 6

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la_m7_handouts - MAC 2103 Module 7 Euclidean Vector Spaces...

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