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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.

User Mike Brady
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1 Answer

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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*1+(-6)*2+3* 0+2*6=1-12+0+12==1\\eq 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.

User Xudifsd
by
7.7k points
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