(a) Is x(x ? 1), (2x ? 1)2 , 1 a spanning set for P2? (b) Are the matrices (1 1

Question

(a) Is x(x ? 1), (2x ? 1)2 , 1 a spanning set for P2?

(b) Are the matrices (1 1 1 1) , ( 0 ?1 1 0 ) , (2 0 0 2) , ( 0 1 ?1 ?1) linearly independent?

(c) It can be shown that the solutions to the differential equation y ?? = y are exactly all functions of the form y = aex + be?x , for constants a and b. (And it is not all that hard, either, but you do not need to do it.) These solutions form a vector space S = Span(e x , e?x ). Show that cosh(x) = e x + e ?x 2 and sinh(x) = e x ? e ?x 2 form a basis for S.

Problem: (a) Is z(z 1), (2z -1)2, 1 a spanning set for P (b) Are the matrices 1 1 (0-1 2 00 1 11"(1 0"(02),(-1-1 linearly independent? (c) It can be shown that the solutions to the differential equation are exactly all functions of the form for constants and b. (And it is not all that hard, either, but you do not need to do it.) These solutions form a vector space Sp Show that form a basis for S

Explanation / Answer

a)

P2 means "up to quadratic" or "up to and including quadritic" polynomials, so I'll address both cases.

If P2 means "up to and including quadratic", then you'll need 3 basis polynomials to span the set (since there are three polynomial degrees), so the set can't span P2.

If P2 means "up to quadratic", then 2+x-x^2 is not in P2, so the given set couldn't possibly span it.

so either way the answer is no.

b).

[1....1]

[1.....1]

det of this matrix = 1-1=0

so this is linearly independent

[0.....-1]

[1......0]

det = 0 +1=1

so this not linearly independent

[2....0]

[0....2]

det = 4-0 =4

so not linearly independent

[0....1]

[-1...-1]

det = 0 +1=1

not linearly independent

post c) and d) in different question

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