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)
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