Outer product vs inner product, and matrix representation of operator

Previously, we have seen that a bra multiply by a ket gives us the inner product, which is a complex number. What happens when we have a ket multiply with a bra? In this case, we shall aptly call this an outer product. Unlike the inner product, the outer product yields a matrix instead. In this video, we introduce the outer product, and from which we derived the completeness relations, and how it can be a very useful tool in deriving the matrix representation of an operator.

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Erratum
0:09:1:25 The complex conjugation should be on the d elements, NOT b. Credit @3721eu

[eu]

I belive you made a mistake in the writing of oter product. If the inner product is <d|b>=((|d>)*)T|b> than the outer product of |b><d| shoud be |b>((|d>)*)T instead of (|b>)*(|d>)T. Or in case you wanted to write |d><b| should have been |d>((|b>)*)T