3) Let V = R[t]<2, W = M2x2 (R) with standard basis B = (1, t, t²), C = (m₁, m2, m3, m4).Suppose : V → W is some linear map such that, under the complexified basis Bc, Cc wehave(a)(b)(c)[yc]c=1 2 00 1 -122 21 4 0What is yc(10 t²)What is p(1+t) (not the complexified function, the original)What is [y]

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Answered: 3) Let V = R[t]≤2, W = M2×2(R) 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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Find a linear mapping G that maps [0, 1] x [0, 1] to the parallelogram in the xy-plane spanned by the vectors (-3,7) and (9,3). (Use symbolic notation and fractions where needed. Give your answer in the form (*, *).) G(u, v) =
V is a 4-dimensional vector space with basis B = (v1, V2 , V3 , V4). T:V → V is linear transformation with rank three and satisfies T(v1) = v + v2 +- v3 + v4, T(v2) = v1 + 2v1 + 3v3 + 4v4, T(v3) = v1 + 3v2 + 4v3 + 6v4. If T(v4) = v1 + 4v2 + 5v3 + avĄ what is the value of a? 8 O 7 O 5
In the vector space P3(R), let S = {p1(x), P2(x), P3(x)}, where p1 (x) = 63x° + 19x² – 37x – 41, P2(x) = –189x – 57x² + 111x + 123, P3 (x) = 45x – 26x? – 55x + 92. (a) Determine whether S is linearly independent. (b) Determine whether (S) = P3(R). (c) Find the dimension of (S). 4.
Consider the basis B₁ = {(1,2), (3, -1)} of R2. Define T: R² = {(1, 0, 0), (1, 1, 0), (1, 1, 1)} of R³ and the basis B₂: R³ such that T((a, b)) = (a + 2b, - You may assume that T is linear. -a, -b). a. Find the matrix M(T) for T with respect to the standard bases in the domain and codomain. b. Find the matrix M(T) for T with respect to B₂ in the domain and B₁ in the codomain.
Given basis B = {(1, -1), (2, 3)} and basis C = {(1, 0), (0, 1)}, and linear map T = (x + 2y ; 2x - y) from R2 --> R2, find the coordinate vectors of basis C in terms of basis B.
Let V = R. Let ß = {(1,1,1), (1, 2, 1), (–1, 1,0)}. (a) Express (x, y, z) as a linear combination of vectors in B. (b) Use (a) to deduce that B is a basis of V. (C) Suppose that T : R³ → R² is a linear map so that T(1, 1, 1) = (2,1), T(1,2, 1) = (–1,3), T(-1,1,0) = (1, –1). Find T(r, y, z). (Hint: Use linearity and (a).)
Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2.r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2πThen the vector-valued functiondescribes a tilted circle (a circle in a plane that is not parallel to one of the coordinateplanes). The center of the circle is O(0, 0, 0).(a) Show that |r(t)| is constant and determine the constant. This is the radius of thecircle.(b) Find an equation for the plane that contains the circle by doing the following:(i) Find three points on the circle.(ii) Use the three points to find a normal vector to the plane containing the three points.(iii) Find an equation for the plane. Check: does the plane contain the center of the circle?(iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)
Let S {V1, V2, V3} and T = {w₁, W2, W3} be two bases in the vector space P₂ {a+bx+cx²: a, b, c = R}, where V₁ = 1+x+3x², v₂ : −2 − 2x − 7x², V3 = 1 + 2x + 6x² and = : 1 + x − x², w₂ = W1 = 1+ 2x², W3 = 3-x+5x2 Find the transition matrix from T to S. Let v = 201 - 3v2 + V3, find the coefficients (C₁, C2, C3) so that v = C₁w₁ + C₂W2 + C3W3. Hint: For the second part, you need also to find the transition matrix from S to T. =
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