100305_ProblemsofIsom

100305_ProblemsofIsom - 4. Show that : defined by + = , is...

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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 : defined by + = , is an isomorphism. Can you find the basis that this isomorphism is based on? 5. Check if the following are isomorphisms: a) , , = + + + + + b) , , = + + c) , , = + , + , + , + + d) , , = + + , + , +...
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