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
$$\begin{pmatrix} x_1^{2} & x_1 & 1 & y_1 \\ x_2^{2} & x_2 & 1 & y_2 \\ x_3^{2} & x_3 & 1 &y_3 \end{pmatrix}$$

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
$$\begin{pmatrix} x_1^{2} & x_1 & 1 \\ x_2^{2} & x_2 & 1 \\ x_3^{2} & x_3 & 1 \end{pmatrix} \begin{pmatrix} a\\ b\\ c\\ \end{pmatrix} = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}$$
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.

 

[blue_raver22]

Hi there,

First of all, congratulations on taking the initiative to teach yourself linear algebra! It's a complex and important subject, and I'm happy to help you with this problem.

To prove that the coefficients a, b, and c of y = ax^2 + bx + c are a solution of the system of linear equations, we can use the augmented matrix provided. The first three columns of the matrix represent the coefficients of x^2, x, and the constant term, respectively. The last column represents the y-values at the points (x1,y1), (x2,y2), and (x3,y3).

To start, let's substitute the values for a, b, and c into the equation y = ax^2 + bx + c. We get:

y1 = a(x1)^2 + b(x1) + c
y2 = a(x2)^2 + b(x2) + c
y3 = a(x3)^2 + b(x3) + c

Now, we can rearrange these equations to match the form of the augmented matrix:

a(x1)^2 + b(x1) + c - y1 = 0
a(x2)^2 + b(x2) + c - y2 = 0
a(x3)^2 + b(x3) + c - y3 = 0

Notice that these equations match the first three rows of the augmented matrix. This means that if we plug in the values for a, b, and c into the matrix, we will get a matrix with all zeros in the last column. This is because the equations are satisfied by the values of a, b, and c, and therefore, the points (x1,y1), (x2,y2), and (x3,y3) lie on the curve y = ax^2 + bx + c.

So, to prove that the coefficients a, b, and c are a solution of the system of linear equations, we simply need to show that the last column of the augmented matrix becomes all zeros when we substitute in the values for a, b, and c. This can be done by row operations on the matrix, such as multiplying a row by a constant or adding/subtracting rows.

I hope this helps clarify how the augmented matrix is related to the curve y = ax^2 + bx + c. Keep up the good work with learning linear algebra!