2. Let V = M₂ (R). Show that the symmetric bilinear form : VxV → R given by (X,Y)= tr(XY)is nondegenerate. (Hint. Suppose X = (ij) is such that Þ(X,Y)= 0 for all Y € V. Plug in Y = ek,the matrix whose (lk)-entry is 1, and all other entries are 0, to show that j = 0 for all i, j.)

Answered: Image /qna-images/answer/a8575de1-7eee-4178-a7a0-114a648cd701.jpg

2. Let V = M₂ (R). Show that the symmetric bilinear form : VxV→R given by (X,Y)= tr(XY) (X,Y)=0 for all Y € V. Plug in Y = ek is nondegenerate. (Hint. Suppose X = (zij) is such that the matrix whose ((k)-entry is 1, and all other entries are 0, to show that zij = 0 for all i, j.)

2. Let V = M₂ (R). Show that the symmetric bilinear form : VxV→R given by (X,Y)= tr(XY) (X,Y)=0 for all Y € V. Plug in Y = ek is nondegenerate. (Hint. Suppose X = (zij) is such that the matrix whose ((k)-entry is 1, and all other entries are 0, to show that zij = 0 for all i, j.)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.8: Determinants

Problem 33E

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Question

2. Let V = M₂ (R). Show that the symmetric bilinear form : VxV → R given by (X,Y)= tr(XY)
is nondegenerate. (Hint. Suppose X = (ij) is such that Þ(X,Y)= 0 for all Y € V. Plug in Y = ek,
the matrix whose (lk)-entry is 1, and all other entries are 0, to show that j = 0 for all i, j.)

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