3). Consider P₁ the space of polynomials of degree less than or equal to 4. Let MC P₁ consis of all polynomials of degree 4 or less which vanish at 1,2,and 3: M = {P(x) = P₁: P(1) = P(2) = P(3)=0}. a) Prove that M is a subspace of P₁. b). What is the dimension of the quotient space P₁/M? c). Find a complimentary to M subspace in P₁.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Solve all parts a,b and c.
3). Consider P4- the space of polynomials of degree less than or equal to
4. Let MC P₁ consis of all polynomials of degree 4 or less which vanish at
1,2,and 3:
M = {P(x) = P₁: P(1) = P(2) = P(3) = 0}.
a) Prove that M is a subspace of P₁.
b). What is the dimension of the quotient space P₁/M?
c). Find a complimentary to M subspace in P4.
Transcribed Image Text:3). Consider P4- the space of polynomials of degree less than or equal to 4. Let MC P₁ consis of all polynomials of degree 4 or less which vanish at 1,2,and 3: M = {P(x) = P₁: P(1) = P(2) = P(3) = 0}. a) Prove that M is a subspace of P₁. b). What is the dimension of the quotient space P₁/M? c). Find a complimentary to M subspace in P4.
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