sa1 - spaces: (a) x 1-x 2 + 4 x 4 + 2 x 5-x 6 = 0 2 x 1-2 x...

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Self-Assessment 1 – Math 5A, Winter 2008 Answer the following questions without looking in the book. If you do not feel comfortable doing this, read the corresponding sections in the book, and then solve the problems, again without looking in the book. 1. State the definition of a vector space. 2. Prove that the set V = { f : R R | f is continuous } is a vector space. 3. Given a matrix A R n × m , show that the set W = { x R m | Ax = 0 } is a vector space. 4. Consider a linear differential equation of the form y 00 + p ( t ) y 0 + q ( t ) y = 0 , (1) where p and q are given continuous functions defined in R . Show that the set S = { y : R R | y is a solution of (1) } is a vector space of dimension 2. 5. Consider the set W = { ( x, y ) | x 2 + y 2 = 1 } ⊂ R 2 . Is W a vector space? Why? 6. Solve the linear following linear systems and determine their solution
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Unformatted text preview: spaces: (a) x 1-x 2 + 4 x 4 + 2 x 5-x 6 = 0 2 x 1-2 x 2 + x 3 + 2 x 4 + 4 x 5-x 6 = 0 1 2 (b) 2 x 1-2 x 2 + 4 x 3 + 2 x 4 = 0 2 x 1 + x 2 + 7 x 2 + 4 x 4 = 0 x 1-4 x 2-x 3 + 7 x 4 = 0 4 x 1-12 x 2-20 x 4 = 0 (c) 3 x 1 + 6 x 3 + 3 x 4 + 9 x 5 = 0 x 1 + 3 x 2-4 x 3-8 x 4 + 3 x 5 = 0 x 1-6 x 2 + 14 x 3 + 19 x 4 + 3 x 5 = 0 7. Determine the dimension and find a basis for W = { ( x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x 1 + x 3 = 0 , x 2 = x 4 } . 8. Determine the dimension and find a basis for W = { ( x 1 , x 2 , x 3 ) ∈ R 3 | x 1 + x 2 + x 3 = 0 } ....
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This note was uploaded on 12/26/2011 for the course MATH 5A taught by Professor Rickrugangye during the Fall '07 term at UCSB.

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sa1 - spaces: (a) x 1-x 2 + 4 x 4 + 2 x 5-x 6 = 0 2 x 1-2 x...

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