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MidISample copy

# MidISample copy - P(b Let R 3 = a(1 x 2 b x-3 2 | a and b...

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Name and SID: 2 1. Let P be the set of all polynomials with real coefficients. (a) Let Q 3 be the set of all polynomials of degree exactly 3 with real coefficients. Is Q 3 a subspace of

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Unformatted text preview: P ? (b) Let R 3 = { a (1 + x 2 ) + b ( x-3 2 ) | a and b are real numbers . } . Is R 3 a subspace of P ? Name and SID: 3 2. Let f 1 = 1 1 1 1 , f 2 = 1 1 1 , f 3 = 1 1 , f 4 = 1 . Show that f 1 , f 2 , f 3 , f 4 are linearly independent. Name and SID: 4 3. Let S = { (1 , 2 , 3) , (2 , 3 , 4) , (3 , 4 , 5) } . Find the dimension and a basis for span ( S ). Name and SID: 5 4. Deﬁne a function T : R 2 → M 2 × 2 ( R ) as T ( a 1 , a 2 ) = ± a 1 a 2 2 a 2 a 1 ² . (a) Show that T is a linear transformation. (b) Find bases for its range and null spaces. Name and SID: 6 5. Find linear transformations U , T : R 2 → R 2 such that UT = T (the zero transformation) but TU 6 = T ....
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