most 2, and letLet P₂ denote the vector space of polynomials of degree atD: P2 → P2be the transformation that sends a polynomial p(t) = at² + bt + c in P₂ to its derivativep'(t) = 2at+b, that is,D(p) = p'.(a) Prove that D is a linear transformation.(b) Find a basis for the kernel ker(D) of the linear transformation D and compute itsnullity.(c) Find a basis for the image im(D) of the linear transformation D and compute itsrank.(d) Verify that the Rank-Nullity Theorem holds for the linear transformation D.(e) Find the matrix representation of D in the standard basis (1, t, t²) of P2.

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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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Let f(2) = 22² – iz be a complex valued function defined on a complex plane. Writing z = x + iy, find real valued functions u(x, y) and v(x, y) such that f(2) = u(x, y) + iv(x, y). u(x, y) = 2x² + 2y² + x_and v(x, y) = 4xy – y u(x, y) = 2x° + 2y² – x and v(x, y) = 4.xy+ y u(x, y) = 2x? – 2y² + y and v(x, y) = 4xy – x u(x, y) = x² +y +x and v(x, y) = 2xy– y Home End PgUp PrtScn F7 F6 F8 F9
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.)
3) Compute the curl of F= (4 xy) i + (2x² + 2z²) j + (4 yz) k.
Consider a constant vector B e R*. Is the (projection) function F(ā)=| a-B à - dot products – where äeR“ a linear transformation? Justify, always.
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Let P₂ denote the vector space of polynomials of degree at D: P2 P2 be the transformation that sends a polynomial p(t) = at² + bt + c in P₂ to its derivative p'(t) = 2at+b, that is, D(p) = p'. most 2, and let