Exercise 3.2.14. In the planes corresponding to the equations and intersect in a line. What is the projection matrix that projects points in onto that line?
Answer: To find the line of intersection we solve the system of equations where
We use Gaussian elimination, subtracting 1 times row 1 from row 2 to produce the echelon matrix
Since has pivots in columns 1 and 2 we have and as basic variables and as a free variable. Setting from the second row of we have or . From the first row of we then have or . So is a solution to the system, as is any other vector on the line through the origin and .
The projection matrix projecting points in onto the line through 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, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.