Let V be the space of polynomials with real coefficients of degree at most n, and...

Let V be the space of polynomials with real coefficients of degree at most n, and let D be the differentiation operator. Find all eigenvectors of D on V.

Expert Solution

Colby Messinger answered 1 year ago

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 ....

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.

3. We let ??(?) denote the set of all polynomials of degree at most n with...

3. We let ??(?) denote the set of all polynomials of degree at most n with real coefficients. Let ? = {? + ??3 |?, ? ??? ???? ???????}. Prove that T is a vector space using standard addition and scalar multiplication of polynomials in ?3(?).

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!

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....

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

Let V be the vector space of 2 × 2 real matrices and let P2 be...

Let V be the vector space of 2 × 2 real matrices and let P2 be the vector space of polynomials of degree less than or equal to 2. Write down a linear transformation T : V ? P2 with rank 2. You do not need to prove that the function you write down is a linear transformation, but you may want to check this yourself. You do, however, need to prove that your transformation has rank 2.

Let Poly3(x) = polynomials in x of degree at most 2. They form a 3- dimensional...

Let Poly3(x) = polynomials in x of degree at most 2. They form a 3- dimensional space. Express the operator T(p) = p' as a matrix (i) in basis {1, x, x 2 }, (ii) in basis {1, x, 1+x 2 } .

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 a vector space and let U and W be subspaces of V ....

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

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