Question

True or False (linear algebra): Every orthonormal set of 4 vectors in R^4 forms a basis of R^4.

Answer 1

its true for linear

Answer 2

awesome! that's what I was thinking, but I'm not sure what I would use for justification. Do you have an idea?

Answer 3

if you have 4 orthonormal vectors, then they are automatically linearly independent. You can show this by saying:\[c_1q_1+c_2q_2+c_3q_3+c_4q_4=0\] Then taking the inner (or dot) product using every vector to show the constants are really 0. For example if I do the inner product with q1 we get:\[<q_1,c_1q_1+c_2q_2+c_3q_3+c_4q_4>=<q_1,0>\]\[\iff c_1=0\]because the inner product of q1 with itself is 1, and with any other vector in the basis is 0. Repeat this with the other vectors to show each constant is 0, and your done.