- #1

RJLiberator

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## Homework Statement

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].

## Homework Equations

Orthonormality conditions:

|v_i>*|v_j> = 0 if i≠j OR 1 if i=j.

## The Attempt at a Solution

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I don't 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.