100305_ProblemsofIsom

100305_ProblemsofIsom - 4 Show that ±:² ³ → ℝ ´...

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1. Consider the subset S 3x3 ( R ) M 3x3 ( R ) consisting of the symmetric matrices, that is, those which equal their transpose. Show that S 3x3 ( R ) is actually a subspace of M 3x3 ( R ) and then determine the dimension and a basis for this subspace. Construct an isomorphism based on your basis from S 3x3 ( R ) to g G . 2. Show that the polynomials P 1 = 2 - x , P 2 = 1 + x + x 2 , and P 3 = 3 x - 2 x 2 from P 2 are linearly independent. Hence they form a basis of P 2. Construct an isomorphism based on this basis from P 2 to g G . 3. Show that the space generated by the row vectors of a matrix is isomorphic to the space generated by its column vectors.
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Unformatted text preview: 4. Show that ±:² ³ → ℝ ´ defined by ±µ¶ + ·¸¹ = µ¶ − ·,·¹ is an isomorphism. Can you find the basis that this isomorphism is based on? 5. Check if the following are isomorphisms: a) ±µ¶ ³ ,¶ ´ ,¶ º ¹ = ¶ ³ + µ¶ ³ + ¶ ´ ¹¸ + µ¶ ³ + ¶ ´ + ¶ º ¹¸ ´ b) ±µ¶ ³ ,¶ ´ ,¶ º ¹ = ¶ ³ + µ¶ ³ + ¶ ´ ¹¸ c) ±µ¶ ³ ,¶ ´ ,¶ º ¹ = µ¶ ³ + ¶ ´ ,¶ ´ + ¶ º ,¶ º + ¶ ³ ,¶ ³ + ¶ ´ + ¶ º ¹ d) ±µ¶ ³ ,¶ ´ ,¶ º ¹ = µ¶ ³ + ¶ ´ + ¶ º ,¶ ³ + ¶ ´ ,¶ ³ + ¶ ´ ¹...
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