math54 - quiz 6 - solns

math54 - quiz 6 - solns - T rA = rA ± 1 1 ² = r ³ A ± 1...

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Math 54 quiz 6 Solutions 1. For the following questions, T : V W is a linear transformation. (a) (1 point) The nullspace of T is a subspace of what space? It is a subspace of V . (b) (2 points) Explain what it means to say that T is onto. T is onto means that T ( V ) = W , or in other words that for every vector ~w W , there is a vector in V that maps to ~w under T . 2. Consider the vector space M 2 × 2 ( R ) of two-by-two matrices. Define a map T : M 2 × 2 ( R ) R 2 for a matrix A M 2 × 2 ( R ) by T ( A ) = A ± 1 1 ² . Either show that T is a linear transformation or explain why it is not. T is a linear transformation. Let’s check the two axioms. For this, take A,B arbitrary matrices, and r R an arbitrary scalar. T ( A + B ) = ( A + B ) ± 1 1 ² = A ± 1 1 ² + B ± 1 1 ² , and this follows from the distributive property of matrix multiplica- tion. For the second axiom, we have
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Unformatted text preview: T ( rA ) = ( rA ) ± 1 1 ² = r ³ A ± 1 1 ²´ , and this follows from the associative property of matrix and scalar multiplication. 3. (3 points) Suppose you know that a transformation T : R 2 → R 2 does the following things T ± 1 ² = ±-2 5 ² , T ± 1 1 ² = ±-1 1 ² . 1 Find the matrix representation A T of T . The matrix representation of T is given by A T = ± T ± 1 ² T ± 1 ²² . We know the first column, but we need to determine the second. T ± 1 ² = T ³± 1 1 ²-± 1 ²´ = T ± 1 1 ²-T ± 1 ² = ± 1-4 ² , where we used the linearity of T for the second equality. This gives A T = ±-2 1 5-4 ² 2...
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This note was uploaded on 09/17/2011 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at Berkeley.

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math54 - quiz 6 - solns - T rA = rA ± 1 1 ² = r ³ A ± 1...

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