Question

Let L: ?4 ? ?3 be a linear transformation defined as L(p(x)) = -10p' + 5p'' for any p(x) ? ?4. a) Determine Ker(L). b) Find rank(L) and nullity(L). c) Is L one-to-one? d) Find L(x^2 - 9). e) Find the coordinate vector of L(x^2 - 9) with respect to the basis S = {x^2 + x + 1, x + 1, 1} in ?3.

          Let L: ?4 ? ?3 be a linear transformation defined as L(p(x)) = -10p' + 5p'' for any p(x) ? ?4.
a) Determine Ker(L).
b) Find rank(L) and nullity(L).
c) Is L one-to-one?
d) Find L(x^2 - 9).
e) Find the coordinate vector of L(x^2 - 9) with respect to the basis S = {x^2 + x + 1, x + 1, 1} in ?3.

Let L: ?4 ? ?3 be a linear transformation defined as L(p(x)) = -10p' + 5p” for any p(x) ? ?4.
a) Determine Ker(L).
b) Find rank(L) and nullity(L).
c) Is L one-to-one?
d) Find L(x^2 - 9).
e) Find the coordinate vector of L(x^2 - 9) with respect to the basis S = x^2 + x + 1, x + 1, 1 in ?3.

Added by David C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

07/25/2023

Step 1

That is, we need to find all p(x) such that -p'(x) + 5p(x) = 0. Let p(x) = ax^4 + bx^3 + cx^2 + dx + e. Then p'(x) = 4ax^3 + 3bx^2 + 2cx + d. Now we plug this into the equation: - (4ax^3 + 3bx^2 + 2cx + d) + 5(ax^4 + bx^3 + cx^2 + dx + e) = 0 Expanding and Show more…

Show all steps

Thanks for your feedback!

Let L: P4 -> P3 be a linear transformation defined as L(p(x)) = -10p' + 5p'' for any p(x) in P4. a) Determine Ker(L). b) Find rank(L) and nullity(L). c) Is L one-to-one? d) Find L(x^2 - 9). e) Find the coordinate vector of L(x^2 - 9) with respect to the basis S = {x^2 + x + 1, x + 1, 1} in P3.