Consider the following functions. f1(x) = 0, f2(x) = x, f3(x) = ex g(x) = c1f1(x) + c2f2(x) + c3f3(x) Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−[infinity], [infinity]). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.) {c1, c2, c3} = Incorrect: Your answer is incorrect. Determine whether f1, f2, f3 are linearly independent on the interval (−[infinity], [infinity]). linearly dependent linearly independent Correct: Your answer is correct.

a) Only a trivial solution exists.

b) Since there is only a trivial solution, then we infer that the set of functions is linearly independent.

a) According to Linear Algebra, a set of elements is linearly independent if and only if a coefficient at least within a linear combination under the following form is not zero:

(1)

Where:

If we know that , and , then we have the following expression:

Given that , we conclude that can be any real number and we can reduce (1) by Compatibility with Addition:

is a trascendental function bounded between 0 and and represents a polynomic function with all real numbers as domain. There is a trivial solution for , but nontrivial ones for all . Hence, we have the following finding:

, and have a set of the form for all .

Now if we consider and represents any real number, then we reduce (1) into this:

And the only choice to meet this linear combination is considering that .

Hence, we have a trivial solution.

b) In consequence, the three functions are linearly independent since it is guaranteed that trivial solution is the only existing solution.

We kindly invite to see this question on linear independence: link

Linearly independent

Step-by-step explanation:

The given functions are:

We want to solve for and  so that on the interval .

Let us substitute to get:

This implies that:

--->eqn1

Let us differentiate each ter wrt x to get:

---->eqn2

Make the subject in equation (2) to get:

---->eqn3

Substitute into equation (1) to get:

Factor nicely to get:

Note that we are after constants that will this equation true for all x. The only way this could be possible is when

Put into eqn3 to get:

Let us substitute everything into g(x) to find

This implies that:

By the zero product principle , for r , we must have either or .

Since is true, is also true

Since  for all x, if , the functions are linearly independent.

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step-by-step explanation:

you bought 31 v bucks

step-by-step explanation:

25+6=31