Consider the vector space P2 of all polynomials of degree less than or equal to 2...

Consider the vector space P2 of all polynomials of degree less than or equal to 2 i.e. P = p(x) = ax + bx + c | a,b,c €.R Determine whether each of the parts a) and b) defines a subspace in P2 ? Explain your answer. a) ( 10 pts. ) p(0) + p(1) = 1 b) ( 10 pts.) p(1) = − p(−1)

Expert Solution

Colby Messinger answered 4 months ago

Let P2 be the vector space of all polynomials of degree less than or equal to...

Let P2 be the vector space of all polynomials of degree less than or equal to 2. (i) Show that {x + 1, x2 + x, x − 1} is a basis for P2. (ii) Define a transformation L from P2 into P2 by: L(f) = (xf)' . In other words, L acts on the polynomial f(x) by first multiplying the function by x, then differentiating. The result is another polynomial in P2. Prove that L is a linear transformation....

4. Whether P3 or the space of the polynomials of degree less than or equal to...

4. Whether P3 or the space of the polynomials of degree less than or equal to 3 and consider T: P3 → P3, given by the derivation T(f) = f' . For example, T (−3x 2 + 5x - 10) = −6x + 5. (a) Prove that T is a linear transformation. (b) Determine ker (T). (c) Is the T transformation injective? Justify that. (d) The polynomial g (x) = 3x^2 + 1 belongs to the image? Justify that.

Let V be the 3-dimensional vector space of all polynomials of order less than or equal...

Let V be the 3-dimensional vector space of all polynomials of order less than or equal to 2 with real coeﬃcients. (a) Show that the function B: V ×V →R given by B(f,g) = f(−1)g(−1) + f(0)g(0) + f(1)g(1) is an inner product and write out its Gram matrix with respect to the basis (1,t,t2). DO NOT COPY YOUR SOLUTION FROM OTHER SOLUTIONS

Recall P2(t) is the set of polynomials of order less than or equal to 2. Consider...

Recall P2(t) is the set of polynomials of order less than or equal to 2. Consider the the set of vectors in P2(t). B={t^2,(t−1)^2,(t+1)^2} (a) Show B is a basis for P2(t). (b) If E={1,t,t^2}is the standard basis, calculate the change of basis matrices PE→B and PB→E (c) Given v= 2t^2−5t+ 3, find its components in B

Let PN denote the vector space of all polynomials of degree N or less, with real...

Let PN denote the vector space of all polynomials of degree N or less, with real coefficients. Let the linear transformation: T: P3 --> P1 be the second derivative. Is T onto? Explain. Is T one-to-one? What is the Kernel of T? Find the standard matrix A for the linear transformation T. Let B= {x+1 , x-1 , x2+x , x3+x2 } be a basis for P3 ; and F={ x+2 , x-3 } be a basis for P1 ....

Verify all axioms that show that the set of second degree polynomials is a vector space....

Verify all axioms that show that the set of second degree polynomials is a vector space. What is the Rank? P2 = {p(x)P | p(x) = ax^2 + bx + c where a,b,c E R}

S_3 is the vector space of polynomials degree <= 3. V is a subspace of poly's...

S_3 is the vector space of polynomials degree <= 3. V is a subspace of poly's s(t) so that s(0) = s(1) = 0. The inner product for two poly. s(t) and f(t) is def.: (s,f) = ([integral from 0 to 1] s(t)f(t)dt). I would like guidance finding (1) an orthogonal basis for V and (2) the projection for s(t) = 1 - t + 2t^2. Thank you!

1. Show that the set of all polynomials of deg=2 is not a vector space over...

1. Show that the set of all polynomials of deg=2 is not a vector space over reals. can this be fixed, can we have a set of polynomials that is a vector space over reals? 2. Show that the set of 2x2 matrices with m_22 = 1 is not a vector space over reals. 3. Show that the set of infinitely-differentiable real functions is a a vector space under pointwise function addition, and pointwise scalar multiplication as defined in class,...

Which of the following sets of polynomials form a basis for P2 (the space of polynomials...

Which of the following sets of polynomials form a basis for P2 (the space of polynomials of degree at most 2)? Explain. (a) {2 + 2x − x 2 , 1 + x, 3x} (b) {1 − x + 2x 2 , 2 + 4x} 1 (c) {3 + x + x 2 , −1 + x, 5 − x + x 2}

Let P denote the vector space of all polynomials with real coefficients and Pn be the...

Let P denote the vector space of all polynomials with real coefficients and Pn be the set of all polynomials in p with degree <= n. a) Show that Pn is a vector subspace of P. b) Show that {1,x,x2,...,xn} is a basis for Pn.

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