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Linear Algebra With Applications
Found in: Page 393
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 a symmetric matrix A. If the vector v is in the image of A and w is in the kernel of A, is v necessarily orthogonal to w? Justify your answer.

The v is orthogonal to w.

See the step by step solution

Step by Step Solution

Step 1: The matrix

  • A matrix is a rectangular array or table of numbers, symbols, or expressions that are organised in rows and columns to represent a mathematical object or an attribute of such an item in mathematics.
  • For example, is a two-row, three-column matrix

Step 2: Determine the orthogonal matrix

A theorem we studied in this book states that

im A-=ker AT

And we know that A is symmetric A=AT, which implies

im A-=ker ATim A-=kerA

Hence v is orthogonal to w.

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