Science:Math Exam Resources/Courses/MATH221/December 2011/Question 06 (a)/Solution 1

First we must determine whether the vectors are orthogonal. We know that two vectors are orthogonal if their dot product is zero.

Consider $v_1\cdot v_2$. This is equal to

$(1/2)(1/2)+(1/2)(1/2)+(1/2)(-1/2)+(1/2)(-1/2)=0.$

All other dot products are calculated in the same way and all vanish. Hence $\{v_1,v_2,v_3,v_4\}$ is an orthogonal set.

Since an orthogonal set of nonzero vectors must be linearly independent, and a set of 4 linearly independent vectors in ${\mathbb{ℝ}}^4$ must be a basis of ${\mathbb{ℝ}}^4$, we conclude that $\{v_1,v_2,v_3,v_4\}$ is an orthogonal basis of ${\mathbb{ℝ}}^4$.