Assume that [itex](|v_1>, |v_2>, |v_3>)[/itex] is an orthonormal basis for V. Show that any vector in V which is orthogonal to [itex]|v_3>[/itex] can be expressed as a linear combination of [itex]|v_1>[/itex] and [itex]|v_2>[/itex].
|v_i>*|v_j> = 0 if i≠j OR 1 if i=j.
The Attempt at a Solution
I dont know how to mathematically prove this obvious one.
I understand the definitions here.
Orthonormal basis implies that the set of three vectors lives in dimension three and are all orthogonal to one another. This means the |v_1> *|v_2> = 0.
If some vector is orthogonal to |v_3> that means |x> |v_3> = 0.
I am pretty sure we have to use sum notation for an inner product and orthonormality conditions to prove this statement.
Hints on starting out properly?
Here's the only lead I have:
|x> = c*|v_1> + t*|v_2>
as it is a linear combination of v_1 and v_2. But we don't want to assume the proof.