Let , and let . Then are orthogonal and neither are zero. Did you mean to assume that ?
Let S = { } be an orthogonal set of vectors in . If S is linearly dependent, prove that one of the must be the zero vector.
Find an orthonormal basis for the column space of the matrix A =
[2 5 7]
[3 1 8]
[6 6 10]
[0 6 -9] and obtain the QR factorisation of A.
That set is not linearly dependent.
wopashui, if the set is linearly dependent, then there exist numbers, , not all 0, such that . Now take the dot product of both sides of that with , , etc.
As for the second problem, think of the three columns of A as three vectors and use "Gram-Schmidt" to find an orthonormal basis.