la_m6_handouts

la_m6_handouts - MAC 2103 Module 6 Euclidean Vector Spaces...

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1 1 MAC 2103 Module 6 Euclidean Vector Spaces I 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Use vector notation in n . 2. Find the inner product of two vectors in n . 3. Find the norm of a vector and the distance between two vectors in n . 4. Express a linear system in n in dot product form. 5. Find the standard matrix of a linear transformation from n to m . 6. Use linear transformations such as reflections, projections, and rotations. 7. Use the composition of two or more linear transformations . http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.
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2 3 Rev.09 Euclidean Vector Spaces I http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Euclidean n -Space -Space , , n Linear Transformations from n to m There are two major topics in this module: 4 Rev.F09 Some Important Properties of Vector Operations in n http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. If u , v , and w are vectors in n and k and s are scalars, then the following hold: (See Theorem 4.1.1) a) u + v = v + u b) u + ( v + w ) = ( u + v ) + w c) u + 0 = 0 + u = u d) u + ( - u) = 0 e) k ( s u ) =( ks) u f) k ( u + v ) = k u + k v g) ( k + s ) u = k u + s u h) 1 u = u
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3 5 Rev.F09 Basic Vector Operations in n http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Two vectors u = ( u 1 , u 2 , , u n ) and v = ( v 1 , v 2 , , v n ) are equal if and only if u 1 = v 1 , u 2 = v , … , u n = v . Thus, u + v = ( u 1 + v 1 , u 2 + v 2 ,…, u n + v n ) u - v = ( u 1 - v 1 , u 2 - v 2 ,…, u n - v n ) and 5 v - 2 u = (5 u 1 - 2 v 1 , 5 u 2 - 2 v 2 ,…, 5 u n - 2 v n ) 6 Rev.F09 How to Find the Inner Product of Two Vectors in n ? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The inner product of two vectors u = ( u 1 ,u 2 ,…,u n ) and v = ( v 1 ,v 2 ,…,v n ), u · v , in n is also known as the Euclidean inner product or dot product. • The inner product , u · v , can be computed as follows: Example: Find the Euclidean inner product of u and v in 4 , if u = (2, -3, 6, 1) and v = (1, 9, -2, 4). Solution: ! u · ! v = u 1 ! v 1 + u 2 ! v 2 + ... + u n ! v n = ! u T ! v ! u · ! v = (2)(1) + ( ! 3)(9) + (6)( ! 2) + (1)(4) = ! 33 = 2 ! 3 6 1 " # $ % 1 9 ! 2 4 " # & & & & $ % ' ' ' '
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4 7 Rev.F09 How to Find the Norm of a Vector in n ? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. As we have learned in a previous module, the norm of a vector in 2 and 3 can be obtained by taking the square root of the sum of square of the components as follows: ! u = u 1 2 + u 2 2 , ! u = ( u 1 , u 2 ) = u 1 u 2 ! " # # $ % & & ! u = u 1 2 + u 2 2 + u 3 2 , ! u = ( u 1 , u 2 , u 3 ) = u 1 u 2 u 3 ! " # # # $ % & & & 8 Rev.F09 How to Find the Norm of a Vector in n ? (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Similarly, the Euclidean norm of u = ( u 1 ,u 2 ,…,u n ), || u || , in n can be computed as follows: Example: Find the Euclidean norm of u = (2, -3, 6, 1) in 4 . Solution: ! u = 2 2 + ( ! 3) 2 + 6 2 + 1 2 = 4 + 9 + 36 + 1 = 50 = 5 2 ! u = 1 2 1 + u 2 2 + ... + u n 2 , !
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la_m6_handouts - MAC 2103 Module 6 Euclidean Vector Spaces...

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