Let Q be a square n xx n matrix. Let {e_1,e_2,...,e_n} be the n standard basis column vectors of RR^n. Show that the set of vectors {Qe_1,Qe_2,...,Qe_n} also form a set of orthonormal vectors.

Cindy Noble 2022-09-06 Answered
Here's the question: Let Q be a square n × n matrix. Let { e 1 , e 2 , . . . , e n } be the n standard basis column vectors of Rn. Show that the set of vectors { Q e 1 , Q e 2 , . . . , Q e n } also form a set of orthonormal vectors.
In terms of my attempts, I've proven that each column vector of Q forms a set of orthonormal vectors in R n . I feel like this may be very close but I'm struggling to picture where to go from here. If this method is correct, where do I go from here? If this method is not correct, what would be the best way to prove this?
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Answers (1)

Yuliana Griffith
Answered 2022-09-07 Author has 6 answers
We have e i T e j = δ i j and want ( Q e i ) T Q e j = δ i j , but the left-hand side is e i T Q T Q e j = δ i j , which is equivalent to Q T Q = I. In particular, if Q T Q e j isn't proportional to e j , some e i , i j won't be orthogonal to it; whereas if Q T Q e j e j , we need equality so e i T Q T Q e j is the identity matrix rather than just being diagonal
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