Exercise 3.4.26. In the Gram-Schmidt orthogonalization process the third component is computed as . Verify that is orthogonal to both and .

Answer: Taking the dot product of and we have

Since and are scalars and and are orthonormal we then have

So is orthogonal to .

Taking the dot product of and we have

So is also orthogonal to .

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fifth Edition and the accompanying free online course, and Dr Strang’s other books.