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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