082nd_tuthome

082nd_tuthome - R 3 = Span v ⊕ ker L 3 Let L V → V be a...

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MATH1111/2008-09/ Take-home Tutorial 1 MATH1111 Tutorial (Take-home) Instruction. Not to be handed 1. Let L : V W be a linear transformation from the vector space V to the vector space W . Prove that dim ker L + dim L ( V ) = dim V where ker L is the kernel of L and L ( V ) is the range of L . 2. Define the linear functional L : R 3 R by L (( x y z ) T ) = x + y + z . (a) Find a v R 3 such that L ( v ) = 1. (b) Find a basis for ker L . (c) Figure out geometrically Span( v ) and ker L , and explain the (geometric) meaning of
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Unformatted text preview: R 3 = Span( v ) ⊕ ker( L ) . 3. Let L : V → V be a linear operator. (a) Show that ker( L ) is an invariant subspace of V under L . (b) Is it possible to find a basis B such that the matrix representation of L w.r.t. B is of the form A = * * ··· * . . . . . . . . . * * ··· * ! ? If yes, how many zero columns does A have?...
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