answer part e (a) Given u =(3,-1,2,3) and v= (0,2,2,1) in R¹. Find 3u-2v.=(4,4,5) as a linear combination of u, = (1,2,3), u, = (-2,0,1) and u, = (1,0,0).(b) Write v=(4,4,5)(c) Determine whether the set R², with the operations(x₁, y₁) + (x₂, y₂ ) = (x,,0) and c(x, y) = (x,cy)is a vector space. If it is, verify each vector space axiom. If not, state all vector space axioms thatfail.(d) Determine whether the set W = {(x₁,x₂,4): x, and x₂ are real numbers is a subspace of R³with the standard operations. Justify your answer.(e) Determine whether the set S = {x²-2x,x² +8,x¹-x²₁x² -4} spans P₁.

Answered: (e) Determine whether the set S = {x² -…

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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Related questions

Question

answer part e

(a) Given u =(3,-1,2,3) and v= (0,2,2,1) in R¹. Find 3u-2v.
=(4,4,5) as a linear combination of u, = (1,2,3), u, = (-2,0,1) and u, = (1,0,0).
(b) Write v=(4,4,5)
(c) Determine whether the set R², with the operations
(x₁, y₁) + (x₂, y₂ ) = (x,,0) and c(x, y) = (x,cy)
is a vector space. If it is, verify each vector space axiom. If not, state all vector space axioms that
fail.
(d) Determine whether the set W = {(x₁,x₂,4): x, and x₂ are real numbers is a subspace of R³
with the standard operations. Justify your answer.
(e) Determine whether the set S = {x²-2x,x² +8,x¹-x²₁x² -4} spans P₁.

Expert Solution

Step 1

To find- Determine whether the set $S = \{x^{2} - 2 x, x^{3} + 8, x^{3} - x^{2}, x^{2} - 4\}$ spans $P_{3}$ .

Concept Used-

The set $S = \{p_{1}, p_{2}, p_{3}, p_{4}\}$ spans $V = P_{3}$ if $a_{1} p_{1} + a_{2} p_{2} + a_{3} p_{3} + a_{4} p_{4} = b_{1} q_{1} + b_{2} q_{2} + b_{3} q_{3} + b_{4} q_{4}$ has at least one solution for every set of values of the coefficient $b_{1}, b_{2}, b_{3}, b_{4}$ .