09-22-10 - 2 Determine whether each of the following are...

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Math 54 Discussion Section Problems Mike hartglass September 22, 2010 You should work on the following problems in groups of 3 or 4. Try to get through as many as you can, but you aren’t expected to finish everything. In fact, the answers are largely unimportant; making sure everyone in your group knows how to solve all the problems is what really matters. 1. Determine which of the following are vector spaces. For those that are, prove it and determine what 0 is. For those that aren’t, show which axiom it violates: (a) The set of all polynomials of degree at most 2 (b) The set of all polynomials of degree exactly 2 (c) The set of all infinite sequences of real numbers (d) The set of all power series (e) The set of all 2 × 3 matrices (f) The set of all continuous functions from R to R 2. Determine whether each of the following are subspaces of
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Unformatted text preview: 2. Determine whether each of the following are subspaces of P 3 1 (a) { p ∈ P 3 : p (1) = 0 } (b) { p ∈ P 3 : p (1) = 1 } (c) The set of all odd functions in P 3 3. Do the invertible 3 × 3 matrices form a subspace of M 3 × 3 ? 2 What about the matrices of the form a b c d e f ? Of the form a b c 1 d e 1 1 f ? 4. Consider the transformation T : P 3 → P 3 given by T ( p ) = p . Prove that T is a linear transformation and determine if it is 1-1 and if it is onto. 5. Let A be some fixed 2 × 3 matrix. Prove that the set { B ∈ M 4 × 2 : AB = 0 } is a subspace of M 4 × 2 . 1 P n denotes the set of all polynomials with degree less than or equal to n 2 M m × n denotes the set of all m × n matrices...
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