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Linear Algebra With Applications
Found in: Page 1
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Consider the function T(A)(X)=xTAx from Rnxn to Qn . Show that T is a linear transformation. Find the image, kernel, rank, and nullity of T.

the solution is


See the step by step solution

Step by Step Solution

Step 1: Given information

TAx=xT Ax

Step 2: Linear transformation and image of T and kernel of T

Consider A,BRnxn and αR,then


For α=1, we get TA+B=TA+TB and for B=0, we get TαA=αTA. So,T:RNXNQn,TAx=xTAx is a linear transformation. Image of T

We know that for any symmetric matrix ARnxntimes n , xTAx, creates a quadratic form(unique), and that the image of T defines the entire space of quadratic form Qn described by the symmetric matrices.


Kernel of T: if A is a skew symmetric matrix, that is AT=-A then ,

xTAx=xTATx=-xTAxBut, since xTAx is a real number, so xTAxT=xTAx and so, we get

xTAx=-xTAxxTAx=0. So, the space of all skew-symmetric matrices is the subset of the kernel of T. By rank-nullity theorem, dimension of kernel of T= nullity of T dimRnxn=n2and rankT=nn+12 As a result, because is the dimension of the space of skew-symmetric matrices, all skew-symmetric matrices make up the whole kernel of T.



Step 3: Conclusion


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