Linear Algebra Solution

We look at the solution to the last problem given:

Find the associated matrix of the quadratic form:

f(X) = x^2 - 3xy + 4y^2

if X = (x, y, z)

Solution:

Let V = R^3 and let t^X = (x,y,z) denote elements of R^3. We have C =

(a.......b)
(c.......d)

and we require: f(x,y, z) = (x,y,z) C *v

where, v =

[x]
[y]
[z]

and x^2 - 3xy + 4y^2 = ax^2 + 2bxy + dy^2 + ez^2

but e = 0 and hence z = 0, so f(x,y,z) reduces to f(x,y) format let t^X = (x,y) denote elements of R^2 such that:

x^2 - 3xy + 4y^2 = ax^2 + 2bxy + dy^2

and: a = 1, 2b = -3 so b = -3/2 and d = 1

Then: C

=(1......-3/2)
(-3/2...... 4)