Linear Algebra and Its Applications, Exercise 3.4.9

Exercise 3.4.9. Given the three orthonormal vectors q_1, q_2, and q_3, what linear combination of q_1 and q_2 is the least distance from q_3?

Answer: Any linear combination of q_1 and q_2 is in the plane formed by q_1 and q_2. The combination closest to q_3 is simply the projection of q_3 onto that plane. But because q_3 is orthogonal to both q_1 and q_2 it is orthogonal to that plane, and its projection onto the plane is the zero vector. So the linear combination of q_1 and q_2 closest to q_3 is 0 \cdot q_1 + 0 \cdot q_2.

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.

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