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 $\mathbb {R} ^{4}$.