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Chapter 5: Q7E (page 245)

Consider a symmetric n脳mmatrix A. What is the relationship between Im(A)and ker(A)?

Short Answer

Expert verified

The relation betweenimAandkerA鈯�isimA=kerA鈯�.

Step by step solution

01

Determine the relation.

Consider a n脳mmatrix A .

Theorem: For any matrix B ,imB=kerB鈯�.

By the theorem, im(A)=kerB鈯�.

02

Draw the graph of relation between Im(A) and ker (A) .

Sketch the graph to show the relation betweenimAandkerA鈯�

Hence, the relation betweenimAandkerA鈯�isimA=kerA鈯�.

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If is a symmetric matrix, then must be symmetric as well.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

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