Question

Consider the following. {(1, −6, 3, 2), (1, 2, 0, 6)}
(a) Determine whether the set of vectors in Rn is orthogonal.
(b) If the set is orthogonal, then determine whether it is also orthonormal.
(c) Determine whether the set is a basis for Rn.

Answer 1

Answer:  The required answers are

(a) NOT orthogonal.

(b) NOT orthonormal.

(c) Not a basis,

Step-by-step explanation:  We are given to consider the vectors {(1, −6, 3, 2), (1, 2, 0, 6)}.

(a) We are to determine whether the set of vectors in $\mathbb{R}^n$ is orthogonal.

(b) If the set is orthogonal, then to determine whether it is also orthonormal.

(c) To determine whether the set is a basis for $\mathbb{R}^n$.

We know that

any two vectors u and v are orthogonal, if their dot product u.v is zero.

The dot product of (1, −6, 3, 2) and (1, 2, 0, 6) is given by

$(1,-6,3,2).(1,2,0,6)=1\times1+(-6)\times2+3\times 0+2\times6=1-12+0+12==1\neq 0.$

So, the given set is not orthogonal.

Since the vectors are not orthogonal, they cannot be orthonormal.

To be a basis of $\mathbb{R}^n$, the set must contain n vectors.

Here, we are dealing with $\mathbb{R}^4,$ and the set contains only 2 vectors.

So, the given set cannot be a basis.