Show that B = {1+ 2t, t – t2,t+t2} is a basis for P3. ) Let p(t) = 1+3t +t. Compute [p(t)]s- (a) (b)

Q: If {e, xe", e2 is a basis of solutions of y" + ay" + by + cy = 0, %3D then a =

A:

Q: Apply the Gram-Schmidt process to find an orthogonal basis for S. 3 S = span 4

A: In this question, we find the orthogonal set of basis by the Gram-Schmidt process.

Q: Let f E P(R) be a polynomial of degree n. Let yo,..., Yn be the dual basis of 1, x, x²,..., x". Show…

A:

Q: Find the coordinate vector of p (x) = 1 + 2x + 3x2   with respect to the basis ẞ = {l + x, 1 -x, x2}…

A: Here given

Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², -1 – 3x + 4x²} O is linearly dependent. O s…

A:

Q: Find a basis for null(T), where T : R^3 → R^2 is defined by T (x, y, z) = (−7x + 2y − 5z, 4x + 9y +…

A:

Q: ,T If {v1, ..., Vn} is an orthonormal basis for R", then vịv¡ + · · ·+ vn Vn = I. True False

A:

Q: Y1 Y2 ER3 [Y3. (x, y) 2x1y1 + X2Y2+5x3Y3, for x = |X3 (a) Find a basis for the orthogonal complement…

A:

Q: 2

A:

Q: Explain why S is not a basis for P2.S = {1 − x, 1 − x2, −1 − 2x + 3x2}

A:

Q: -2 Let T : R? → R² be defined by Tx = Ax where A = Find a basis 10. B for R² such that the B-matrix…

A:

Q: 3) Let S contain vectors of the form f(x) = a + bx + cx² + dx³ : f(-2) = 0 and f'(-4)= 0: Then a…

A:

Q: ii) Show that S = {(t – 1)³, (t – 1)², (t – 1), 1} is a basis of P3(t) where P3(t) stands for the…

A:

Q: Let S = {V1, V2} be a basis of R? where v1 and v2 = -3 What element of R? is determined by the…

A:

Q: 2. Let T: P2- P2 be defined by T(a+ bx + c?) = a - 2b+ (2b – 4c)x+3ca². Find a basis C for P2 such…

A: Given T:P2→P2 defined as…

Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², –1 – 4x + 5x²} S is linearly dependent. S…

A: Given a set S={1-x, 1-x2, -1-4x+5x2}  To Explain why S is not a basis for P2.

Q: 1. Find [p]a, relative to the basis 3 = {P1. P2: Ps} for P, where p = 4- 3x +x², P1 = 1, P2 = X, P3…

A: Given

Q: (b) Let W be the eigenspace of B corresponding to A = 1. Find a basis for W.

A:

Q: Find a basis for Col B where [1 1 0 -2 0 0]

A:

Q: Extend {l + x, 1 + x + x2} to a basis for P 2.

A:

Q: 2. Let T : P2 –→ P2 be defined by T(a + bx + cx²) = a – 26 + (2b – 4c)x + 3 cx?. Find a basis C for…

A:

Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², –1 – 6x + 7x²} O s is linearly dependent. O…

A:

Q: now that B = {x, r³ + 2x² + 4, x² + 2, 1} is a basis for P3.

A:

Q: Explain why S is not a basis for P,. .2 S = (1 - x, 1 - x², -1 – 8x + 9x²} O s is linearly…

A:

Q: Define a basis v1, v2 of R² by vị obtained by applying the Gram-Schmidt Procedure to v1, v2? 2 by…

A: Consider the provided question, A Basis of a vector space. Let V be a subspace of Rn for some n. A…

Q: Let B = {1+t,1 – t} and C = {4, 1 + 5t, t²}. Does B form a basis for P2?

A:

Q: Let s = {(r² –3r+4, 2r-5, x²+1} be a basis for P2, let q E P2. If (q). = [2,-3, 1], then q = %3D %3D…

A:

Q: 4.13 Let {u1,u2,..., un} be any orthonormal basis of (where F R or F = C). Show that %3D Euju; = 1.…

A:

Q: In the vector space P of polynomials of degree atmost two, the third column of the matrix of…

A: The solution is given below in the next step:

Q: Let V = span {1+ 2x – x² + 3r³, -2+x+x² – x*, -1+ 8r – x² + 7r", 3 + x – 2r² + r'}. Find a basis ß…

A: Step:-1 Note:- To span V we need the polynomials of degree 3, so dimension of V is 4. (think using…

Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², -1 – 6x + 7x2} || O sis linearly dependent.…

A:

Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², -1 – 6x + 7x²} O sis linearly dependent. O…

A:

Q: Q: Find a representation of Jp(t)> at a time t +0 using the ordered basis {|e1>, ]e2>, ]e3> }.

A:

Q: Explain why S is not a basis for P. S = {1, 2x, =7 + x², 9x} O S is linearly dependent.

A:

Q: Show that B = {1,1 – x,2 – 4x + x²,6 – 18x + 9x² – x³} is a basis for P3(R). Hint: Recall that…

A:

Q: 3 Find the coordinate vector of x = -2 with respect to the basis B for R3. -2 6 ||

A: Given that the vector x→=3-2-2 and the basis is 166,01-5,001. It is required to find the coordinate…

Q: Find a basis for the plane 4x + 3y + 5z = 0 in R³.

A:

Q: 6. Define fe (R2)* by f(x, y) 2x + y and T: R2 (3x + 2y, x). (a) Compute T'(f). (b) Compute [T']p,…

A: We will use the basic knowledge of linear transformations to answer all the three given subparts of…

Q: The Gram-Schmidt process t ransforms the basis U13D(-1,3), u2=(7,-1) of R into the orthogonal basis…

A: Option (E) is correct.

Q: Explain why S is not a basis for P2. S = {1, 8x, -6 + x², 3x} O sis linearly dependent. O s does not…

A:

Q: 2. Construct a basis for T2,2(R²) and prove that it is a basis.

A: To find the basis of I2×2ℝ2. The set of all 2×2 matrices over ℝ. Let us claim that…

Q: find basis and the 1) dimension of w ={A ¢ Mnz: A +=0}

A:

Q: 9. Let i, j denote the standard basis in R2. Find a scalar c if the vector 7i- 2j is perpendicular…

A: Dot product of the two vectors.

Q: 1. Find [p]8, relative to the basis B = {p1, P2, P3} for P2 where p = 4 – 3x + x², p1 = 1, p2 = X,…

A: Q1. p = 4 - 3x + x² = (4)p1 + (-3)p2 + (1)p3  So, [p]β = [4   -3   1]

Q: Let P2 be the polynomials of degree no more than 2. Is {x2+x+1,2x+1,3x2+1} a basis for P2 ?

A:

Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x2, -1 – 7x + 8x²} O sis linearly dependent. O…

A:

Q: Let T: P2 that is, 7. > P2 be the linear operator T(p(x)) = p(2x+ 1), T(co+ C,x+ C,x²) = c, + c,(2x…

A:

Q: Suppose T E L(V, W), and w1, ..., wm is a basis of range T.

A: Given that T∈L(V,W)andw1,.......wm is a basis of range T.To prove there exist…

Q: a b 1) Let T cla + 2b + 1c + 3d + (-11) e+ (-la+ (-1) b + (-2) c + Od + 7e) x. d to Then a basis for…

A: The given transformation is  Tabcde=1a+2b+1c+3d+-11e+-1a+-1b+-2c+0d+7ex To find a basis for KerT⊥.

Q: 10)Define TE L(R²) by T(x, y) = (41x + 7y,-20x + 74y). Verify that the basis of T %3D with respect…

A: Question 10. The detailed solution is as follows below: