Exercise 3.3.10. Given mutually orthogonal vectors , , and and the matrix with columns and , what are and ? What is the projection of onto the plane formed by and ?
Answer: We have
where the zero entries are the result of and being orthogonal.
Similarly we have
where the zero entries are the result of and being orthogonal to .
Since is orthogonal to both and it is orthogonal to any linear combination of and and therefore is orthogonal to the plane spanned by and . The projection of onto that plane is therefore the zero vector.
This also follows from the formula for the projection matrix corresponding to the matrix . The projection of onto the column space of (the space spanned by and ) is
since as discussed above.
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, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.