Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
2. Let V = Mn(R). Show that the symmetric bilinear form : V xV →R given by (X, Y) = tr(XY) is nondegenerate. (Hint. Suppose X = (xij) is such that (X,Y) = 0 for all Y E V. Plug in Y = eek, the matrix whose (lk)-entry is 1, and all other entries are 0, to show that xij = 0 for all i, j.)

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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 10AEXP
See similar textbooks
icon
Related questions
Question
100%
2. Let V = Mn (R). Show that the symmetric bilinear form : V xV → R given by (X, Y) = tr(XY) is nondegenerate. (Hint. Suppose X = (xij) is such that (X,Y) = 0 for all Y E V. Plug in Y = eek; the matrix whose (lk)-entry is 1, and all other entries are 0, to show that rij O for all i, j.)
Transcribed Image Text:2. Let V = Mn (R). Show that the symmetric bilinear form : V xV → R given by (X, Y) = tr(XY) is nondegenerate. (Hint. Suppose X = (xij) is such that (X,Y) = 0 for all Y E V. Plug in Y = eek; the matrix whose (lk)-entry is 1, and all other entries are 0, to show that rij O for all i, j.)
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer