Suppose that u1 u2 um are nonzero pairwise orthogonal ve

Suppose that {u1, u2, . . . , um} are non-zero pairwise orthogonal vectors (i.e., uj · ui = 0 if i 6= j) of a subspace W of dimension n. Show that m n.

Solution

Given u_{1 ,} u_{2 , ----------} u_m are ortogonal vectors.

the set of orthogonal vectors are linearly independent------------(1)

to prove the above statement true ,let us assume that u_i, u_{j are not linearly independent}

_then u_{i =} k u _{j .hence} u_i . u_j = k u _j . u_j which is not 0. hence statement (1) is true

given W is a vector space of dimension \'n\'. which implies

W has maximum number of linearly independent vectors ------------(2)From statement 1 and 2 it is clear that the number of elements in the given set (m) is less than or equql to the number of elements in the basis of the vector space W (n)

therefore\' m\' is less than or equal to \' n\'.