Get the answer to "Let V=Span{f1,f2,f3}, where f1=1,f2=ex,f3=xex a) Prove that S={f1,f2,f3}..."

College
Algebra
Linear algebra
Vectors and spaces

licencegpopc

Answered question

2022-01-24

Let

V = S p a n {f_{1}, f_{2}, f_{3}}

, where

f_{1} = 1, f_{2} = e^{x}, f_{3} = x e^{x}

a) Prove that

S = {f_{1}, f_{2}, f_{3}}

is a basis of V. b) Find the coordinates of

g = 3 + (1 + 2 x) e^{x}

with respect to S. c) Is

{f_{1}, f_{2}, f_{3}}

linearly independent?

Answer & Explanation

KickAntitte06

Beginner2022-01-25Added 11 answers

Explanation:
V is a vector space. A vector space is defined as the set of all possible linear combination of its basis vectors, where the coefficients are taken from some fields K.
So, if our space V has basis

{v_{1}, v_{2}, \dots, v_{n}}

, and K=R, then a generic element

v \in V

can be written as

v = a_{1} v_{1} + a_{2} v_{2} \dots + a_{n} v_{n}

, where

a_{1}, \dots, a_{n} \in R

In your case, the vector space is defined as the span, i.e. all the possible linear combinations, of the three functions

f_{1} = 1, f_{2} = e^{x}

and

f_{3} = x e^{x}

This means that every element in your space V has the form

a_{1} \cdot 1 + a_{2} \cdot e^{x} + a_{3} \cdot x e^{x} = a_{1} + a_{2} \cdot e^{x} + a_{3} \cdot x e^{x}

So, for example,

5 - 3 e^{x} + 18 x e^{x}

is an element of your space.

Sean Becker

Beginner2022-01-26Added 16 answers

Point one
From all weve

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