Exercise 3.4.12. Given the vectors and , find a scalar such that is orthogonal to . Given the matrix whose columns are and respectively, find matrices and such that is orthogonal and .
Answer: We must have . This implies that or
So the scalar multiplying is 2.
As a check, we then have
with . So the new vector is orthogonal to .
We now attempt to factor into . If we perform Gram-Schmidt orthogonalization on , the first column of is . We have , so . The length would then be the 1, 1 entry of the matrix . We thus have
The second column of is created by first subtracting from the projection of onto and then normalizing the result. The result of subtracting the projection is
The length of this vector is , so we have
The matrix is then
The second diagonal entry of the matrix is the length used in computing . The off-diagonal element of is the value used in subtracting from the component in the direction of . We thus have
The product of the two matrices is then
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.