mat67-Homework3

mat67-Homework3 - MAT067 University of California, Davis...

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MAT067 University of California, Davis Winter 2007 Homework Set 3: Exercises on Linear Spans and Bases Directions : Please work on all exercises! Hand in Problems 1 and 2 as your ”Calculational Homework” and Problems 5 and 7 as your ”Proof-Writing Homework” at the beginning of lecture on January 26, 2007. As usual, we are using F to denote either R or C . 1. Show that the vectors v 1 = (1 , 1 , 1), v 2 = (1 , 2 , 3), and v 3 = (2 , - 1 , 1) are linearly independent in R 3 . Write the vector v = (1 , - 2 , 5) as a linear combination of v 1 , v 2 , and v 3 . 2. Consider the complex vector space V = C 3 and the list ( v 1 , v 2 , v 3 ) of vectors in V , where v 1 = ( i, 0 , 0), v 2 = ( i, 1 , 0), and v 3 = ( i, i, - 1). Show that span( v 1 , v 2 , v 3 ) = V . 3. Find a basis for the subspace U of R 5 defined by U = { ( x 1 , x 2 , . . . , x 5 ) | x 1 = 3 x 2 , x 3 = 7 x 4 } . 4. Let V be a vector space over F , and suppose that the list ( v 1 , v 2 , . . . , v n ) of vectors spans V , where each
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