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2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг...
Let V be the vector space of all polynomials of degree at most 2 equipped with...
Let V be the vector space of all polynomials of degree at most 2 equipped with the inner product defined by (p,q) = p(-1)q (-1) + p (0)g(0) +p(1)q(1),p(x),g(x) E V Find a nonzero polynomial that is orthogonal to both p(x) = 1 + x + x2, and q(x) = 1-2x + x2
let P3 denote the vector space of polynomials of degree 3 or less, with an inner product defined by 14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product def...
let P3 denote the vector space of polynomials of degree 3 or
less, with an inner product defined by
14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements
14. Let Ps denote the vector space of polynomials of degree 3 or less,...
Let P2 be the real vector space of polynomials in a of degree at most 2, and let T be the real vector space of upper tr...
Let P2 be the real vector space of polynomials in a of degree at most 2, and let T be the real vector space of upper triangular 2 x 2 matrica b,cERThe vector space P2 is equipped with the inner product 〈p(x), q(x)-1 p(z)q(z) dr, and the vector space T is equipped with the inner product 〈A.B)=tr(AB), where tr denotes trace. Let L: P2→T be 1.p(z)dr]. Find L 0 c given by L(p(z)):-17(1) .CE :J ) 1 2 0 p(-1)...
Q3. Consider the vector space P, consisting of all polynomials of degree at most two together...
Q3. Consider the vector space P, consisting of all polynomials of degree at most two together with the zero polynomial. Let S = {p.(t), p2(t)} be a set of polynomials in P, where: pi(t) = -4 +5, po(t) = -3° - 34+5 (a) Determine whether the set S = {P1(t).pz(t)} is linearly independent in Py? Provide a clear justification for your solution. (8 pts) (b) Determine whether the set S = {p(t),p2(t)} spans the vector space P ? Provide a...
E the vector space of polynomials over R of degree less than 3. Define a quadratic form on V by a...
e the vector space of polynomials over R of degree less than 3. Define a quadratic form on V by a) Find the symmetric bilinear forma f such that q(p) = f(p, p). b) Consider the basis oy-(1,2-x U)o. c) Let R-(3,2-r, 4-2z +2.2} of V. Find the matrix {f}3: You may give your ,24 of V. Find the matrix answer as a product of matrices and/or their inverses.
e the vector space of polynomials over R of degree less...
Let V = R3[x] be the vector space of all polynomials with real coefficients and degress not exceeding 3. Let V-R3r] be t...
Let V = R3[x] be the vector
space of all polynomials with real coefficients and degress not
exceeding 3.
Let V-R3r] be the vector space of all polynomials with real coefficients and degress not exceeding 3. For 0Sn 3, define the maps dn p(x)HP(x) do where we adopt the convention thatp(x). Also define f V -V to be the linear map dro (a) Show that for O S n 3, T, is in the dual space V (b) LetTOs Show...
Q2 (10 points) Let V- Ps be the vector space of polynomials of degree 3. Let...
Q2 (10 points) Let V- Ps be the vector space of polynomials of degree 3. Let C (1,x 2, 2)2 +2)3) be two ordered bases of V. () Find the change-of-basis matrices Pc-B and PB-c (ii) Find [y]в if [v]c- (1, 0, 0, 1). (iii) Find [y]c if [y]B-( 1, 0, 0, 1).
Given the vector space R[2]deg<s of polynomials with real coefficients of degree at most 5, and...
Given the vector space R[2]deg<s of polynomials with real coefficients of degree at most 5, and Ui = {p(z) : p(z) a? + bz5, for abe R}, find a subspace U2 such that R deg< 5 = Ui φ Ủy Is this U2 unique? (g) If V be a finite dimensional vector space, dim V = n and B = 〈ui,u2, . . . , un) is a basis of V, then show that:
Let V be the vector space of the polynomials of K [t] of degree less than 3, that is, the form p (x) = a2t2 + a1t1 + a0t0. Investigate the linear independence of the polynomials: p1 (t) = 1t2 + 0t1 +...
Let V be the vector space of the polynomials of K [t] of degree
less than 3, that is, the form p (x) = a2t2 +
a1t1 + a0t0.
Investigate the linear independence of the polynomials:
p1 (t) = 1t2 + 0t1 +
1t0, p2 (t) = 2t2 + 2t1
+ 0t0, p3 (t) = 4t2 +
1t1 + 3t0 for:
*b) The modulo operation on 5
*c) The modulo operation on 7
a) K-R b) K c) K
a)...
7. Let V = Pa(R), the vector space of polynomials over R of degree less than 2, with inner product Define φ E p by φ(g)-g(-1) a) By direct calculation, find f e V such that (S)-dg). You are given...
7. Let V = Pa(R), the vector space of polynomials over R of degree less than 2, with inner product Define φ E p by φ(g)-g(-1) a) By direct calculation, find f e V such that (S)-dg). You are given that A 1, V3-2v) is an orthonormal basis for V (you do not need to check this). b) Find the same f as in part a, using the formula for A(6) from class.
7. Let V = Pa(R), the vector...
Let V be the vector space of all polynomials of degree at most 2 equipped with the inner product defined by (p,q) = p(-1)q (-1) + p (0)g(0) +p(1)q(1),p(x),g(x) E V Find a nonzero polynomial that is orthogonal to both p(x) = 1 + x + x2, and q(x) = 1-2x + x2
let P3 denote the vector space of polynomials of degree 3 or
less, with an inner product defined by
14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements
14. Let Ps denote the vector space of polynomials of degree 3 or less,...
Let P2 be the real vector space of polynomials in a of degree at most 2, and let T be the real vector space of upper triangular 2 x 2 matrica b,cERThe vector space P2 is equipped with the inner product 〈p(x), q(x)-1 p(z)q(z) dr, and the vector space T is equipped with the inner product 〈A.B)=tr(AB), where tr denotes trace. Let L: P2→T be 1.p(z)dr]. Find L 0 c given by L(p(z)):-17(1) .CE :J ) 1 2 0 p(-1)...
Q3. Consider the vector space P, consisting of all polynomials of degree at most two together with the zero polynomial. Let S = {p.(t), p2(t)} be a set of polynomials in P, where: pi(t) = -4 +5, po(t) = -3° - 34+5 (a) Determine whether the set S = {P1(t).pz(t)} is linearly independent in Py? Provide a clear justification for your solution. (8 pts) (b) Determine whether the set S = {p(t),p2(t)} spans the vector space P ? Provide a...
e the vector space of polynomials over R of degree less than 3. Define a quadratic form on V by a) Find the symmetric bilinear forma f such that q(p) = f(p, p). b) Consider the basis oy-(1,2-x U)o. c) Let R-(3,2-r, 4-2z +2.2} of V. Find the matrix {f}3: You may give your ,24 of V. Find the matrix answer as a product of matrices and/or their inverses.
e the vector space of polynomials over R of degree less...
Let V = R3[x] be the vector
space of all polynomials with real coefficients and degress not
exceeding 3.
Let V-R3r] be the vector space of all polynomials with real coefficients and degress not exceeding 3. For 0Sn 3, define the maps dn p(x)HP(x) do where we adopt the convention thatp(x). Also define f V -V to be the linear map dro (a) Show that for O S n 3, T, is in the dual space V (b) LetTOs Show...
Q2 (10 points) Let V- Ps be the vector space of polynomials of degree 3. Let C (1,x 2, 2)2 +2)3) be two ordered bases of V. () Find the change-of-basis matrices Pc-B and PB-c (ii) Find [y]в if [v]c- (1, 0, 0, 1). (iii) Find [y]c if [y]B-( 1, 0, 0, 1).
Given the vector space R[2]deg<s of polynomials with real coefficients of degree at most 5, and Ui = {p(z) : p(z) a? + bz5, for abe R}, find a subspace U2 such that R deg< 5 = Ui φ Ủy Is this U2 unique? (g) If V be a finite dimensional vector space, dim V = n and B = 〈ui,u2, . . . , un) is a basis of V, then show that:
Let V be the vector space of the polynomials of K [t] of degree
less than 3, that is, the form p (x) = a2t2 +
a1t1 + a0t0.
Investigate the linear independence of the polynomials:
p1 (t) = 1t2 + 0t1 +
1t0, p2 (t) = 2t2 + 2t1
+ 0t0, p3 (t) = 4t2 +
1t1 + 3t0 for:
*b) The modulo operation on 5
*c) The modulo operation on 7
a) K-R b) K c) K
a)...
7. Let V = Pa(R), the vector space of polynomials over R of degree less than 2, with inner product Define φ E p by φ(g)-g(-1) a) By direct calculation, find f e V such that (S)-dg). You are given that A 1, V3-2v) is an orthonormal basis for V (you do not need to check this). b) Find the same f as in part a, using the formula for A(6) from class.
7. Let V = Pa(R), the vector...
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