Linear Algebra Notes and Confusion.

Question

Hi, while reading the beginning of the notes in my linear algebra section I immediately hit a wall of confusion. (I've been googling on this and can't find out how he did this.) In the notes (which I will post after this explanation) he shows a set of points. Maybe I'm forgetting something from past math classes, or maybe I missed something, but I don't understand how he goes from Q=(1,-6) to knowing all of the coefficients in front of each x. Can someone help me?

Notes:

#4 Examples of questions involving linear equations and more vector geometry details.

(a) Determine the quadratic (parabola) that passes through the points P=(1,-6), Q=(3,4) and R=(-2,9)

(b) Determine all the quadratics (parabolas) that pass through the points (1,1) and (3,5)

Now the form of a quadratic is usually expressed as ax² + bx +c = q(x) , for this problem the unknowns will be

a, b and c. The first step: determine the equations to be solved , the second step: convert matrix form, third step:

solve the matrix form, fourth step: give the conclusion about the quadratic involved.

(a) Here q(1) = -6 , gives a + b + c = -6, q(3) = 4 gives 9a + 3b + c = 4, and q(-2) = 9 gives 4a -2b +c = 9.

Now it helps to briefly examine at these equations before just setting up the matrix form. For these equations the

unknown c seems easier to deal with than a and b, so that suggests reversing the equations to

c + b + a = -6 , c +3b + 9a = 4 , c -2b +4a = 9 now time to convert to a matrix and find its RREF

Answer 1

So the point is that you want to find a parabola that passes through the three points. Luckily for us, when you plug in the x and y values for each point into the general quadratic equation, you get a system of three equations of just a, b and c. The point of linear algebra is to use matrices to solve such systems.

Using the three equations you gave we can create the matrix

c b a

[ 1 1 1 | -6

1 3 9 | 4

1 -2 4 | 9]

From here you can row reduce the matrix using Gaussian elimination to find the exact values of c, b and a. Let me know if you need any more help, I love tutoring Linear Algebra and I can help you with the class.