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Kunze Linear Algebra
· Problem 10.2.1
Bilinear Forms
Mathematics — Methods · quadratic-forms-symmetric-bilinear
Problem
The following expressions define quadratic forms
q
q
q
on
R
2
R^2
R
2
. Find the symmetric bilinear form
f
f
f
corresponding to each
q
q
q
. (a)
a
x
1
2
ax_1^2
a
x
1
2
. (b)
b
x
1
x
2
bx_1 x_2
b
x
1
x
2
. (c)
c
x
2
2
cx_2^2
c
x
2
2
. (d)
2
x
1
2
−
1
3
x
1
x
2
2x_1^2 - \frac{1}{3}x_1 x_2
2
x
1
2
−
3
1
x
1
x
2
. (e)
x
1
2
+
9
x
2
2
x_1^2 + 9x_2^2
x
1
2
+
9
x
2
2
. (f)
3
x
1
x
2
−
x
2
2
3x_1 x_2 - x_2^2
3
x
1
x
2
−
x
2
2
. (g)
4
x
1
2
+
6
x
1
x
2
−
3
x
2
2
4x_1^2 + 6x_1 x_2 - 3x_2^2
4
x
1
2
+
6
x
1
x
2
−
3
x
2
2
.
Related theory
Binary Quadratic Forms Introduction
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