Use an augmented matrix to prove

[John O' Meara]

Hi,
I have just started teaching linear algebra to myself. I know nothing about linear algebra so if this question seems simple please bare with me.
What do I do to show that the coefficients, a,b, and c of y=ax^2+bx+c are a solution of the system of linear equations whose augmented matrix is
$$\begin{pmatrix}<br /> x_1^{2} & x_1 & 1 & y_1 \\<br /> x_2^{2} & x_2 & 1 & y_2 \\<br /> x_3^{2} & x_3 & 1 &y_3 \end{pmatrix}$$

Where the points (x1,y1), (x2,y2) and (x3,y3) are three separate points on the curve y. As a matter of fact I am trying to envisage the three linear equations and how they are related to the curve y. Thanks. The title is not accurate.

 

[Zaphos]

Maybe it'll be more clear to look at the non-augmented system first
$$\begin{pmatrix}<br /> x_1^{2} & x_1 & 1 \\<br /> x_2^{2} & x_2 & 1 \\<br /> x_3^{2} & x_3 & 1 \end{pmatrix}<br /> \begin{pmatrix}<br /> a\\<br /> b\\<br /> c\\ \end{pmatrix} = \begin{pmatrix}<br /> y_1 \\<br /> y_2 \\<br /> y_3 \end{pmatrix}$$
Multiply out a row of it, symbolically, and you can see you're expressing that curve equation y=ax^2+bx+c in matrix form, except with x,y as fixed values and a,b,c as free variables.

So, any a,b,c which satisfies the first row must correspond to a curve going through your first point, (x1,y1). Likewise with the second row for the second point, and third row for third point.

 

[John O' Meara]

I am only 6 pages into the linear algebra book, it started off with linear equations and an augmented matrix, it has not said anything up to that on any other type of matrix or how to multiply a matrix. I can now see from your reply how the expression for the curve y=ax^2+bx+c in matric form arises, just muliply each row's element by a,b,c respectively. However I still do not 'get' the augmented matrix. Thanks for replying.

 

[Zaphos]

The augmented matrix is just a different notation for writing a matrix equation like Ax=b -- smoosh A and b together, and you have the augmented matrix representing Ax=b. This notation is convenient for algorithms like Gaussian elimination, but conceptually I find it nicer to look at the non-augmented form.