HW2 - C[0 1 define f g by f g x = f x g x(we know that...

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Math 601 HOMEWORK 2 1. Show that the set of 2 × 2 matrices with real entries, with usual addition and multiplication with scalars, is a real vector space. Find its dimension and a basis. 2. For each n denote by P n the vector space of polynomials of degree at most n , with coefficients in R : P n = { p ( t ) = a 0 + a 1 t + a 2 t 2 + . . . + a n t n | a j R } . Consider the following linear transformations: S : P 3 → P 4 defined by S ( p ) = p 0 (0) T : P 3 → P 4 defined by T ( p ) = ( x + 2) p ( x ) H : P 4 → P 3 defined by H ( p ) = p 0 ( x ) + p 0 (0) a) Give the formula for S + T , and for 2 T . b) Give the formula for H T . What is its domain? c) Find the matrix representation of H in the basis { 1 , x, x 2 , x 3 , x 4 } of P 4 and { 1 , x, x 2 , x 3 } of P 3 . d) Prove that T is one to one, but not onto. e) Prove that H is onto, but not one to one. 3. Consider the set of continuous functions on the interval [0 , 1]: C [0 , 1] = { f : [0 , 1] R | f continuous } with the natural addition: for
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Unformatted text preview: C [0 , 1] define f + g by ( f + g )( x ) = f ( x )+ g ( x ) (we know that then f + g is also continuous) and natural multiplication by scalars: if c ∈ R and f ∈ C [0 , 1] then cf ∈ C [0 , 1] is defined as ( cf )( x ) = cf ( x ). a) Show that C [0 , 1] is a vector space over R . What is its dimension? b) (BONUS extra 10p.) Show that the functions e x , e 2 x , e 3 x are linearly independent in C [0 , 1]. c) Assuming b) is true, let V = Sp ( e x , e 2 x , e 3 x ) subspace of C [0 , 1], and let T : V → V be given by T ( p ) = p ( x ). Show that T is invertible and find T-1 . ( Hint: find T-1 on a basis, then extend by linearity to its domain V .) 1...
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