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Chapter 5: Q26E (page 263)

TRUE OR FALSE

26. If Vis a subspace of Rnand x鈫�is a vector in Rn, then vector projVx鈫�must be orthogonal to vector x鈫�-projVx鈫�.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Step-by-step solution Step 1: Determine the definition of an orthogonal matrix

A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

Or it can be said that, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

02

Consider an example

For example,

Suppose A is a square matrix with real elements and of n脳norder and AT is the transpose of A. Then according to the definition, if, AT=A-1is satisfied, then,

A路AT=I

Because the projVx鈫� component is taken out of vector x.

Thus, vector projVx鈫� must be orthogonal to vector x鈫�-projVx鈫�.

Hence, the statement is true.

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Most popular questions from this chapter

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