Let $U = <(1,3,-5),(2,0,-4)>$. Find an orthogonal basis of U that contains the vector $(2,0,-4)$. So I did the scalar product of the basis of $U$ and equaled it to zero and got the condition $(2z,z,z)$. How do I include the vector in this basis?
Orthogonal basis of a subspace that contains a vector of that subspace
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linear-algebra
orthogonality
1 Answers
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You already have $\;(2,0,-4)\;$ given as part of a basis of $\;U\;$ , so apply directly to that basis Gram-Schmidt:
$$u_1:=\frac{(2,0,-4)}{\left\|(2,0,-4)\right\|}=\frac1{2\sqrt5}(2,0,-4)$$
and now
$$w_2:=(1,3,-5)-\left((1,3,-5)\cdot u_1\right)u_1\;,\;\;\text{and define}\;\;u_2:=\frac{w_2}{\left\|w_2\right\|}$$
Prove now that $\;\{u_1,\,u_2\}\;$ is an orthonormal basis of your subspace