I'm having trouble getting this started.
Let A1,...,Ar be vectors in R^n and assume that theu are mutually perpendicular (i.e. any two of them are perpendicular), and that none of them are equal to 0. Prove that they are linearly independent.
I'm having trouble getting this started.
Let A1,...,Ar be vectors in R^n and assume that theu are mutually perpendicular (i.e. any two of them are perpendicular), and that none of them are equal to 0. Prove that they are linearly independent.
You have to show that if $\displaystyle x_1A_1+\ldots+x_rA_r = 0$, then all the coefficients $\displaystyle x_1,\ldots,x_r$ must be zero.
Take the scalar product of both sides of that equation with A_j, where j is any one of the numbers from 1 to r: $\displaystyle (x_1A_1+\ldots+x_rA_r)\cdot A_j = 0\cdot A_j$. Obviously the right-hand side of that equation is 0. What can you say about the left-hand side?