1.Show that (u; v) = u1v1 + 2u2v2 + 3u3v3 is an inner product on R3 for
u = (u1; u2; u3) and v = (v1; v2; v3).
2. Let u = (1; 1; 1); v = (1; 1; 0) and w = (1; 0; 0). Show that B = (u; v; w) is linearly independent and
spans R3
3. Transform B into an orthonormal basis using the inner product in 1.
4. Let R3 have the Euclidean inner product and W = span (u; v) where
u =(4/5;0;3/5) and v = (0; 1; 0):
Express w = (1; 2; 3) in the form w = w1 + w2 where w1 is an element of W and w2 is an element of W?

Question

1.Show that (u; v) = u1v1 + 2u2v2 + 3u3v3 is an inner product on R3 for 
u = (u1; u2; u3) and v = (v1; v2; v3). 
2. Let u = (1; 1; 1); v = (1; 1; 0) and w = (1; 0; 0). Show that B = (u; v; w)  is linearly independent and
spans R3
3.  Transform B into an orthonormal basis using the inner product in 1.
4. Let R3 have the Euclidean inner product and W = span (u; v) where
u =(4/5;0;3/5) and v = (0; 1; 0):
Express w = (1; 2; 3) in the form w = w1 + w2 where w1 is an element of  W and w2 is an element of W?

turingtest · Answer

what are the requirements of an inner product?

turingtest · Answer

@romeo.nkala I can't help you if you leave when I ask a question

anonymous · Answer

ow sorry about that..

anonymous · Answer

its the 4 axioms

turingtest · Answer

1) we need to show that$$<\vec u,\vec v>=<\vec v,\vec u>$$since this we know that$$<\vec u,\vec v>= u_1v_1 + 2u_2v_2 + 3u_3v_3$$it follows from the fact that scalar multiplication is commutative that$$ <\vec v,\vec u>=v_1u_1 + 2v_2u_2 + 3v_3u_3=<\vec u,\vec v>$$so that's theorem one down..