Find the orthogonal canonical reduction of the quadratic form −x^2 +y^2 +z^2 −6xy−6xz+2yz. Also, find its principal axes, rank and signature of the quadratic form.

Question

this question is from Linear Algebra.

Answer 1

4.4

Systems of Equations - Three Variables

Objective: Solve systems of equations with three variables

using addi-

tion/elimination.

Solving systems of equations with 3 variables is very simila

r to how we solve sys-

tems with two variables. When we had two variables we reduced

the system down

to one with only one variable (by substitution or addition).

With three variables

we will reduce the system down to one with two variables (usua

lly by addition),

which we can then solve by either addition or substitution.

To reduce from three variables down to two it is very importan

t to keep the work

organized. We will use addition with two equations to elimin

ate one variable.

This new equation we will call (A). Then we will use a different

pair of equations

and use addition to eliminate the

same

variable. This second new equation we

will call (B). Once we have done this we will have two equation

s (A) and (B)

with the same two variables that we can solve using either met

hod. This is shown

in the following examples.

Example 1.

3

x

+ 2

y

−

z

=

−

1

−

2

x

−

2

y

+ 3

z

= 5

We will eliminate

y

using two different pairs of equations

5

x

+ 2

y

−

z

=