A3 - M ◦ L is one-to-one(b Give an example where M is not...

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Math 235 Assignment 3 Due: Wednesday, May 25th 1. Prove that any plane through the origin in R 3 is isomorphic to R 2 . 2. For each of the following pairs of vector spaces, define an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) R 2 and P 1 . b) The vector space S = { A M 2 × 2 ( R ) | A 1 1 0 1 = 1 1 0 1 A } and the vector space U = { p ( x ) P 2 | p (1) = 0 } . 3. In each of the following cases, determine whether h , i defines an inner product on P 2 . (a) h p, q i = p (0) q (0) + p (1) q (1) (b) h p, q i = p ( - 1) q ( - 1) + 2 p (0) q (0) + p (1) q (1) (c) h p, q i = p ( - 1) q (1) + 2 p (0) q (0) + p (1) q ( - 1) 4. Let L : V U and M : U W be linear mappings. (a) Prove that if L and M are one-to-one, then M L
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Unformatted text preview: M ◦ L is one-to-one. (b) Give an example where M is not one-to-one, but M ◦ L is one-to-one. (c) Is it possible to give an example where L is not one-to-one, but M ◦ L is one-to-one? Explain. 5. Let V and W be vector spaces with dim V = n and dim W = m , let L : V → W be a linear mapping, and let A be the matrix of L with respect to bases B for V and C for W . a) Define an explicit isomorphism from Range( L ) to Col( A ). Prove that your map is an isomorphism. b) Use a) to prove that rank( L ) = rank( A )....
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