Vector, u:=[u_1,…,u_n]T^. Find a coordinate transformation matrix, Q in R^(nxn), which is nonsingular, satisfying: [0,...,0,||u||]=Qu

Celinamg8 2022-09-20 Answered
Vector, u := [ u 1 , , u n ] T . I am trying to find a coordinate transformation matrix, Q R n × n , which is nonsingular, satisfying:
[ 0 0 | | u | | ] = Q u .
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Answers (2)

Abagail Stephenson
Answered 2022-09-21 Author has 9 answers
Recall that if W is a subespace of R n , then
dim ( W ) + dim ( W ) = n
Take W as the span of u, so
dim ( < u > ) = n 1.
You can construct a matrix Q that will satisfy the same condition taking any set of n 1 vectors v i from any basis of the ortogonal complement of the span of u.
Q = ( v 11 v 12 v 1 n v 21 v 22 v 2 n v 31 v 32 v 3 n v n 1 , 1 v n 1 , 2 v n 1 , n u 1 | | u | | u 2 | | u | | u n | | u | | )
where v i = ( v i 1 , , v i n ) T are vectors in a basis { v 1 , , v n 1 } of the ortogonal complement of the linear span of г, for all i with 1 i n 1.
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Darius Miles
Answered 2022-09-22 Author has 3 answers
This matrix Q is non singular because it's rows are linearly independent because the following set is a basis of R n :
B = { u , v 1 , , v n 1 }
Also: You can use Gram Schmidt process to get an ortogonal basis of < u > starting from u. In this way you will find the rights vectors v i such that Q is inversible.
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