Linear independence in polynomials

Question

Are the polynomials $4−6x+10x^2, 4+8x+6x^2\ and\ 8+9x+14x^2$ linearly independent in P2?

If they are linearly dependent, find numbers, not all zero, that make the equation below true. If they are linearly independent, then leave the answers blank.

____ $(4−6x+10x^2)+$ _____ $(4+8x+6x^2)+$ ____ $(8+9x+14x^2)$

I put them in homogeneous system of equations and got the following reduced row echelon form: $$\begin{bmatrix} 1&0&\frac 12 \\ 0&1& \frac 32 \\ 0&0&0\end{bmatrix}$$

I figured its dependent but I cannot do the second part correctly

Answer 1

$$\frac{1}{2}(4−6x+10x^2)+\frac{3}{2}(4+8x+6x^2)-(8+9x+14x^2)=0$$

Coefficients were taken directly from the right column of your matrix.

Answer 2

Let us forget the polynomials for a second and think back to what a matrix is supposed to represent.

If we have $Ax = 0$ and you have found the reduced $A$ to a row echelon form it then its really simple to plug in $x = [a,b,c]^T$ and recover (simplified) set of equations you have started with.

In this case (just calculating out the matrix vector product, it is all) we have: $$Ax = [a +\tfrac{1}{2}c, b + \tfrac{3}{2}c,0]^T = [0,0,0]^T.$$

Can you figure out a,b and c from here?

Hint:

The matrix is clearly singular, therefore there are many solutions. Pick a specific value for $c$ and see if you can find a solution for it.