**Polynomial Vector Space**Let \( V \) be the set of polynomials with real coefficients having degree at most 2. Prove that \( V \) is a vector space over the field of real numbers. Prove that the set \(\{1, x, x^2\} \) is a basis of \( V \) and give matrix representations \( D_1, D_2: V \to V \) of the first and second derivative mappings. Does \( D_2 = D_1 \circ D_1 \)?

let V be the set of polynomials with real coefficients having degree at most 2. to prove- (i) V is a…

Answered: Let V be the set of polynomials with…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Using linear algebra, show that G={p1(x)=1−x+x^2, p2(x)=x, p3(x)=x+x^2} is a basis of P2, the vector space of polynomials of degree at most two with real coefficients.
Consider the operator Son the vector space by S(a+bx) = - a+b+ ( a + 2b) x A) Pick a basis B = { b₁,b₂3. Find the minimal polynomials NT, b, (X), Nr, 0₂ (x), and Ns (x) R₁ [x] given B) Show that S is cyclic by finding a vector v such that
Let W be the subspace of R3 with orthonormal basis {w1, w2}, where
Let w,=(1,0,0,1) and w=(-1,0,0,1). Let W=(w | w w, = w-w, = 0}, where w-w, denotes the dot product of w, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique?
Could you explain how to solve this linear algebra question?
7. Let V be a vector space and suppose B = {u, v} is a basis for V. Prove that B' = {ku, v} is also a basis for V for every nonzero scalar k. (Hint: You must prove that B'is linearly independent and that it spans V).
Consider the vector space V = R 3 over the field of real numbers F = R. Do the following vectors form a basis for R 3 ? , ,
Let fi = 1+2x + 3x², f2 = 2 + 3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao +a1x+azx²[ao – 2a1 +a2 = 0} of R2[r]. Let g = 1+5x + 9x². If [g]g = (,) then s = a) -7 b) -5 c) 5 d) 7 e) g is not in V.
Let P₂ be the vector space of polynomials of degree at most 2. Determine the coordinates of the polynomial p(x) = x+x²-2 relative to the basis B = {1, x -2, x²-x}.
Identify all the polynomial functions of two variables of degree < 2 which are harmonic. Show that they form a vector space of dimension 5 and give a vector basis of that space.
Let W be the subspace spanned by -31 64 3 -3 2 5 2 3 Find an orthonormal basis for W.

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