tut1 - L 1 L 2 and L 3 in L b Calculate L 1 2 L 2-L 3 c...

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Math 235 Tutorial 1 Problems 1: Find a basis for the four fundamental subspaces of A = 1 6 0 2 - 1 0 0 1 - 2 1 0 0 1 - 2 1 0 0 0 0 1 . 2: Let L : P 2 R 2 be defined by L ( a + bx + cx 2 ) = a - b b - c a) Prove that L is linear. b) Evaluate L (1 + x + x 2 ). c) Find a polynomial p ( x ) such that L ( p ( x )) = 3 2 . 3: Let L denote the set of all linear mappings with domain P 2 and codomain M 2 × 2 ( R ). a) Give explicit definitions for three distinct linear mapping
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Unformatted text preview: L 1 , L 2 , and L 3 in L . b) Calculate L 1 + 2 L 2-L 3 . c) Show that c ( L 1 + L 2 ) = cL 1 + cL 2 for any c ∈ R . d) Is { L 1 ,L 2 ,L 3 } linearly independent or linearly dependent? e) Find a linear mapping L ∈ L such that L 6∈ Span { L 1 ,L 2 ,L 3 } . 1...
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