la_m9_handouts - MAC 2103 Module 9 General Vector Spaces II...

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1 1 MAC 2103 Module 9 General Vector Spaces II 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Find the coordinate vector with respect to the standard basis for any vector in ℜⁿ 2. Find the coordinate vector with respect to another basis. 3. Determine the dimension of a vector space V from a basis for V. 4. Find a basis for and the dimension of the null space of A, null(A). 5. Find a basis for and the dimension of the column space of A, col(A). a) Show that the non-leading columns of A are linearly dependent since they can be written as a linear combination of the leading columns of A. b) Show that the leading columns of A are linearly independent and therefore form a basis for col(A). 6. Find a basis for and the dimension of the row space of A, row(A). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.
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2 3 Rev.09 General Vector Spaces II http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Coordinate Vectors Coordinate Vectors Basis and Dimension Basis and Dimension Null Space, Column Space, and Null Space, Column Space, and Row Space Row Space of a Matrix of a Matrix There are three major topics in this module: 4 Rev.F09 Quick Review http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. In module 8, we have learned that if we let S = { v 1 , v 2 , … , v r } be a finite set of non-zero vectors in a vector space V, the vector equation has at least one solution, namely the trivial solution , 0 = k 1 = k 2 = … = k r . If the only solution is the trivial solution, then S is a linearly independent set. Otherwise, S is a linearly dependent set. If every vector in the vector space V can be expressed as a linear combination of the vectors in S, then S is the spanning set of the vector space V. If S is a linearly independent set, then S is a basis for V and span(S) = V. k 1 ! v 1 + k 2 ! v 2 + ... + k r ! v r = k i ! v i i = 1 r ! = ! 0
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3 5 Rev.F09 What is the Coordinate Vector with Respect to the Standard Basis for any Vector? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The set of standard basis vectors in ℜⁿ is B = { e 1 , e 2 , … , e n }. If v ℜⁿ , then and has components The coordinate vector v B has the coefficients from the linear combination of the basis vectors as its components. So, A better name might be coefficient vector, but it is not used. So, v is its own coordinate vector with respect to the standard basis, ! v = v 1 v 2 ... v n ! " # $ T = v 1 ˆ e 1 + v 2 ˆ e 2 + ... + v n ˆ e n v 1 , v 2 ,..., v n . ! v = ! v B . ! v B = v 1 v 2 ... v n ! " # $ T . 6 Rev.F09 What is the Coordinate Vector with Respect to Another Basis? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Example: Let B 1 = { v 1 , v 2 }, with To show B 1 is a basis, we solve If the only solution is , then v 1 , v 2 are linearly independent, and B 1 = { v 1 , v 2 } is a linearly independent set and a basis for ² .
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