Exercise 3.4.19. When doing Gram-Schmidt orthogonalization, an alternative approach to computing 
Show that the second method is equivalent to the first.
Answer: We substitute the expression for 
![c' = c'' - (q_2^Tc'')q_2 = [c - (q_1^Tc)q_1] - \left[q_2^T\left[c - (q_1^Tc)q_1\right]\right]q_2](https://s0.wp.com/latex.php?latex=c%27+%3D+c%27%27+-+%28q_2%5ETc%27%27%29q_2+%3D+%5Bc+-+%28q_1%5ETc%29q_1%5D+-+%5Cleft%5Bq_2%5ET%5Cleft%5Bc+-+%28q_1%5ETc%29q_1%5Cright%5D%5Cright%5Dq_2&bg=ffffff&fg=333333&s=0&c=20201002)
![= c - (q_1^Tc)q_1 - \left[q_2^Tc - q_2^T(q_1^Tc)q_1\right]q_2](https://s0.wp.com/latex.php?latex=%3D+c+-+%28q_1%5ETc%29q_1+-+%5Cleft%5Bq_2%5ETc+-+q_2%5ET%28q_1%5ETc%29q_1%5Cright%5Dq_2&bg=ffffff&fg=333333&s=0&c=20201002)
![= c - (q_1^Tc)q_1 - \left[q_2^Tc - (q_1^Tc)q_2^Tq_1\right]q_2](https://s0.wp.com/latex.php?latex=%3D+c+-+%28q_1%5ETc%29q_1+-+%5Cleft%5Bq_2%5ETc+-%C2%A0%28q_1%5ETc%29q_2%5ETq_1%5Cright%5Dq_2&bg=ffffff&fg=333333&s=0&c=20201002)
![= c - (q_1^Tc)q_1 - \left[q_2^Tc - (q_1^Tc) \cdot 0\right]q_2](https://s0.wp.com/latex.php?latex=%3D+c+-+%28q_1%5ETc%29q_1+-+%5Cleft%5Bq_2%5ETc+-%C2%A0%28q_1%5ETc%29+%5Ccdot+0%5Cright%5Dq_2&bg=ffffff&fg=333333&s=0&c=20201002)
The resulting equation
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition
My Math Homework 