Doubt in exercise 8.2 in Linear Algebra by Hoffman and Kunze

Question

Here is the problem from the book.

Consider $R^4$ with the standard inner product. Let $W$ be the subspace of $R^4$ consisting of all vectors which are orthogonal to both $\alpha = (1, 0, -1, 1)$ and $\beta = (2, 3, -1, 2)$. Find a basis for $W$.

I am trying to solve this problem using Gram-Schmidt orthogonalization process but it isn't working out. Is it possible to solve this using Gram-Schmidt process or there is some other way? In either case, please help me as I am reading the book on my own.

Answer 1

You can write your space as $$W=\{(x,y,z,w)\in\mathbb{R}^4: \langle(x,y,z,w),(1,0,-1,1)\rangle=0,\,\langle(x,y,z,w),(2,3,-1,2)\rangle=0\}.$$ The two inner product conditions give you a (homogeneous) system of linear equations with two free variables, so you should get a 2-dimensional space.

Answer 2

Hint: If you just let a general vector $r=(w,x,y,z)$ be any such vector, then you have 2 equations expressing that $r$ is orthogonal to each of $\alpha,\ \beta$. Work with those equations using say row reduced matrix.

Answer 3

Yes, you can use the Gram-Schmidt procedure. Extend the set of vectors $\ \{\alpha, \beta\}\ $ to a basis of $\ R^4\ $ by adding another two appropriate vectors to it. The vectors $\ \gamma=(0,0,1,0)\ $ and $\ \delta=(0,0,0,1)\ $ will do for this purpose because $\ \alpha, \beta, \gamma, \delta\ $ are linearly independent. Now use the Gram-Schmidt procedure to orthonormalise this basis in the order $\ \alpha, \beta, \gamma, \delta\ $. If the resulting orthonormal basis is $\ u_1,u_2,u_3,u_4\ $, then $\ u_1=\frac{\alpha}{\|\alpha\|}\ $ and $\ u_2=\frac{\|\alpha\|^2\beta-\langle\beta,\alpha\rangle\alpha}{\left\| \|\alpha\|^2\beta-\langle\beta,\alpha\rangle\alpha\right\|}\ $ will span the same space as $\ \alpha\ $ and $\ \beta\ $, and $\ u_3, u_4\ $ will be a basis for $\ W\ $.