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skylit

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Forgive me for not writing in latex, but I searched this site for 10 minutes looking for a latex reference and could not find anything on matrices. Also, excuse for the excessive amount of info.

Determine whether this list of 3 polynomials in P1:

p1 = 1+3x

p2 = 1+2x

p3 = 2+3x

is linearly independent. If not, write one of the pi in terms of the others.

To test independence, let's see if the linear system

x1 p1 + x2 p2 + x3 p3 = 0 has any non-trivial solutions.

First, write the coefficient matrix A for a linear system representing the polynomial equation.

I reduced the matrix

[1 1 2]

[3 2 3]

to

[1 0 -1]

[0 1 -3]

I set Ax=0.

I found that the set {pi} is linearly dependent, and that a non-trivial solution to AX = 0 is (0,0,0) (Is this always the case? Pretty much all of the problems that I've come across have this as a solution for AX=0)

Now, my the next part is my issue. The directions then read:

This is where I am totally lost. I'm solving problems via a website, and every attempt to this solution has returned wrong. I've guessed every combination of p1 = p2 + p3 and I am not sure how to approach this last question.

There are numerous problems that ask the same question listed above. It would be greatly appreciated if someone could give an explanation and the proceeding steps to solving a problem like this as I will have to apply it to other similar problems.

## Homework Statement

Determine whether this list of 3 polynomials in P1:

p1 = 1+3x

p2 = 1+2x

p3 = 2+3x

is linearly independent. If not, write one of the pi in terms of the others.

To test independence, let's see if the linear system

x1 p1 + x2 p2 + x3 p3 = 0 has any non-trivial solutions.

First, write the coefficient matrix A for a linear system representing the polynomial equation.

## Homework Equations

## The Attempt at a Solution

I reduced the matrix

[1 1 2]

[3 2 3]

to

[1 0 -1]

[0 1 -3]

I set Ax=0.

I found that the set {pi} is linearly dependent, and that a non-trivial solution to AX = 0 is (0,0,0) (Is this always the case? Pretty much all of the problems that I've come across have this as a solution for AX=0)

Now, my the next part is my issue. The directions then read:

**"In particular, your solution to AX = 0 implies**

= 0

Now use this linear dependence relation among the vectors { p1, p2, p3 } to write one of these vectors as a linear combination of the others."= 0

Now use this linear dependence relation among the vectors { p1, p2, p3 } to write one of these vectors as a linear combination of the others."

This is where I am totally lost. I'm solving problems via a website, and every attempt to this solution has returned wrong. I've guessed every combination of p1 = p2 + p3 and I am not sure how to approach this last question.

There are numerous problems that ask the same question listed above. It would be greatly appreciated if someone could give an explanation and the proceeding steps to solving a problem like this as I will have to apply it to other similar problems.

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