How do I prove $v_{1}v_{1}^T + v_{2}v_{2}^T + .......+ v_{n}v_{n}^T = I$ if ${v_{1},v_{1},...,v_{n}}$ is an orthonormal basis?

Question

So I am trying to understand the following proof so that I can understand another proof which is the SVD theorem.

I tried to prove this by saying $i,j$ element in the resulting matrix is given by $\sum_{k=1}^n v_{ki}v_{kj}$. However, I am stuck as I can't prove this sum is zero when $i \ne j$ and is 1 when $i = j$. So I hoped for help if possible.

Answer 1

Multiply the left-hand expression by $v_j$ on the right, and check that you get $v_j$. Since the left-hand expression has this property on an entire basis, it must equal $I$ which also has this property on an entire basis.