mat67-Homework2

mat67-Homework2 - U ⊂ R 2 such that U is closed under...

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MAT067 University of California, Davis Winter 2007 Homework Set 2: Exercises on Linear Equations and Vector Spaces Directions : Submit your solutions to Problems 1, 2 and 4. Separately, please also submit the Proof-Writing-Problems 3 and 5. This homework is due on Friday January 19, 2007 at the beginning of lecture. As usual, we are using F to denote either R or C . 1. Solve the following systems of linear equations and characterize their solution set (unique solution, no solution, . ...). Also write each system of linear equations as an equation for a single function f : R n R m for appropriate m, n . (a) System of 3 equations in the unknowns x, y, z, w x + 2 y - 2 z + 3 w = 2 2 x + 4 y - 3 z + 4 w = 5 5 x + 10 y - 8 z + 11 w = 12 . (b) System of 4 equations in the unknowns x, y, z x + 2 y - 3 z = 4 x + 3 y + z = 11 2 x + 5 y - 4 z = 13 2 x + 6 y + 2 z = 22 . (c) System of 3 equations in the unknowns x, y, z x + 2 y - 3 z = - 1 3 x - y + 2 z = 7 5 x + 3 y - 4 z = 2 . 2. Show that the space V = { ( x 1 , x 2 , x 3 ) F 3 | x 1 + 2 x 2 + 2 x 3 = 0 } forms a vector space. 3. Let V be a vector space over F . Then, given a F and v V such that av = 0, prove that either a = 0 or v = 0. 4. Give an example of a nonempty subset
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Unformatted text preview: U ⊂ R 2 such that U is closed under scalar multiplication but is not a subspace of R 2 . 2 5. Let V be a vector space over F , and suppose that W 1 and W 2 are subspaces of V . Prove that their intersection W 1 ∩ W 2 is also a subspace of V . 6. Prove or give a counterexample to the following claim: Claim. Let V be a vector space over F , and suppose that W 1 , W 2 , and W 3 are subspaces of V such that W 1 + W 3 = W 2 + W 3 . Then W 1 = W 2 . 7. Let F [ z ] denote the vector space of all polynomials having coeFcient over F , and de±ne U to be the subspace of F [ z ] given by U = { az 2 + bz 5 | a, b ∈ F } . ²ind a subspace W of F [ z ] such that F [ z ] = U ⊕ W . 8. Prove or give a counterexample to the following claim: Claim. Let V be a vector space over F , and suppose that W 1 , W 2 , and W 3 are subspaces of V such that W 1 ⊕ W 3 = W 2 ⊕ W 3 . Then W 1 = W 2 ....
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mat67-Homework2 - U ⊂ R 2 such that U is closed under...

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