Set Cases
In each of the following two cases, the set of vectors {v1,...,vn} forms and orthogonal basis for the
corresponding Euclidean vector space R^n.
Find an expression for the given vector w in terms of the basis.
(Do not use Gaussian elimination).
(a) n = 2, v1 = [1/sqrt(2) 1/sqrt(2) ]^T, v2 = [-1/sqrt(2) 1/sqrt(2) ]^T,
w = [9 4]^T.
(b) n = 4, v1 = [1/3sqrt(2) 1/3sqrt(2) 4/3sqrt(2) 0]^T , v2 = [1/sqrt(6)
-1/sqrt(6) 0 -2/sqrt(6)]^T, v3 = [ 1/sqrt(3) -1/sqrt(3) 0 1/sqrt(3)]^T
v4 = [2/3 2/3 -1/3 0]^T, w = [4 3 0 -2]^T.
The most elegant way to solve these problems is to construct a change of basis matrix and multiply the provided vector w by this matrix.
A change of basis matrix is easily constructed from the set {v1, ..., vn} of basis vectors. The rows of the matrix are given by the vectors v1^T, v2^T, ..., vn^T, if the vectors v1, v2, ..., vn are column vectors as I suspect they are based on the problem's use of the transpose.
Matrices are difficult to illustrate textually. Besides, you shouldn't have any trouble with this problem given the above procedure.
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