Let VR and suppose p and q are nonnegative integers such that p+q=n. For = (₁₁) and y=(n) in V, set (x, y)p.q=211 + +Ipyp-p+13p+1-nn. Then (): V XV R is a symmetric bilinear form. The notation RP is often used to denote R" equipped with the bilinear form (,)p. For example, if p= n and q = 0, then (,-)n,o is just the usual dot product. The forms (,)3,1 and (,)1,3 are sometimes called the Lorentzian inner product or Minkowski inner product, and play an important role in physics. 1. Let ß (een) be the standard basis of R", i.e. e, has 1 in the i-th place and 0 everywhere else. Calculate the Gram matrix of (,)pg with respect to B and conclude that this bilinear form is nondegenerate.
Let VR and suppose p and q are nonnegative integers such that p+q=n. For = (₁₁) and y=(n) in V, set (x, y)p.q=211 + +Ipyp-p+13p+1-nn. Then (): V XV R is a symmetric bilinear form. The notation RP is often used to denote R" equipped with the bilinear form (,)p. For example, if p= n and q = 0, then (,-)n,o is just the usual dot product. The forms (,)3,1 and (,)1,3 are sometimes called the Lorentzian inner product or Minkowski inner product, and play an important role in physics. 1. Let ß (een) be the standard basis of R", i.e. e, has 1 in the i-th place and 0 everywhere else. Calculate the Gram matrix of (,)pg with respect to B and conclude that this bilinear form is nondegenerate.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let VR and suppose p and q are nonnegative integers such that p+q=n. For z = (₁,..., In)
and y=(₁₁n) in V, set
(z,y)p.q=13/1+
Then
+Ipyp-p+1Yp+1-
-Inn.
(): V XV R
is a symmetric bilinear form. The notation RP is often used to denote R" equipped with the bilinear
form (,)p. For example, if p = n and q = 0, then (,),o is just the usual dot product. The forms
()3,1 and ()1,3 are sometimes called the Lorentzian inner product or Minkowski inner product, and
play an important role in physics.
1. Let B (een) be the standard basis of R", i.e. e, has 1 in the i-th place and 0 everywhere
else. Calculate the Gram matrix of (,)pa with respect to B and conclude that this bilinear form is
nondegenerate.
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