Linear algebra solutions & Matrices of quadratic forms

We look at the solutions from the earlier blog:

1) For a similar solid to that shown, but with vectors:

u = (1, -1, 4)

v = (1, 1, 0)

w = (-1, 2, 5)

Find: Vol (u, v, w)

Solution:

We let the volume be Vol(u, v, w) and:Vol (u, v, w) = Det [u, v, w]

So that:

(u1....u2......u3)
(v1....v2.....v3)
(w1....w2....w3) =

(1.....-1.....4)
(1......1.... .0)
(-1....2.......5)

And Det [u, v, w] =

(1....0)
{2.....5) -

(-1)(1....0)
(-1.....5) +


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

[(5 - 0) + (5 - 0) + 4(2 - (-1)] = 5 + 5 + 4(3) = 10 + 12 = 22 cu. units


2) Show that for the spanning vectors:

u = (-2, 2, 1)

v = (0, 1, 0)

w = (-4, 3, 2)


Vol (u, v, w) = 0

Solution:

Vol (u, v, w) = Det [u, v, w]

So that:

(u1....u2......u3)
(v1....v2.....v3)
(w1....w2....w3) =

(-2.....2......1)
(0.... 1.....0)
(-4....3.......2)

And Det [u, v, w] =

(-2)(1....0)
(3....2) -

2(0....0)
(-4.....2) +

(0.....1)
(-4.....3) =

[-2(2 - 0) - 2(0) + (0 -(-4)] = -4 + 4 = 0

Given these were fairly short, we now make a brief foray into the area of quadratic forms and the matrices associated with them. Let V be a finite dimensional space over the field K. Let g = < , > be a symmetric bilinear form on V. When we say a quadratic form determined by g we mean the function: f:V -> K

such that: f(v) = g =

Generic example:

If V = K^n then f(X) = X*X = (x1)^2 + .........(x_n)^2

i.e. for the quadratic form determined by an ordinary dot product. In general if V = K^n and C is a symmetric matrix in K, representing a bilinear form, the quadratic form is given as a function of X by:

f(X) = t^XCX = (SIGMA) i,j = 1 to n [c_ijxixj]

where SIGMA is for the Greek sumbol of summation, then if C is a diagonal matrix, e.g.

C =

(c1....0 ......0)
(0.....c2......0)
(0......0......c3)

then the quadratic form has the simpler expression:

f(X) = c1x1^2 + ........................cn x_n^2


Specific example - application:

Let V = R^2 and let t^X = (x,y) denote elements of R^2.

Given a function f(x,y) = 2x^2 + 3xy + y^2

is a quadratic form, find the matrix of its bilinear symmetric form.

We have C =

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

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

where, v =

[x]
[y]

then:

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

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

Then: C =

(2........3/2)
(3/2......1)



Problem:

Find the associated matrix of the quadratic form:

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

if X = (x, y, z)