Well I thought the outer product was more complicated than this.

a × bᵀ    instead of    aᵀ  ×  b

An inner product is constructed by multiplying vectors A and B like Aᵀ × B. (ᵀ is for turned.) In other words, timesing each a guy from A by his corresponding b guy from B.

After summing those products, the result is just one number.  In other words the total effect was to convert two length-n vectors into just one number. Thus mapping a large space onto a small space, Rⁿ→R.  Hence inner.

Outer product, you just do × Bᵀ.  That has the effect of filling up a matrix with the contents of every possible multiplicative combination of a's and b's.  Which maps a large space onto a much larger space — maybe squared as large, for instance putting two Rⁿ vectors together into an Rⁿˣⁿ matrix.

a × bᵀ    instead of    aᵀ  ×  b

No operation was done to consolidate them, rather they were left as individual pieces.

So the inner product gives a “brief” answer (two vectors ↦ a number), and the outer product gives a “longwinded” answer (two vectors ↦ a matrix). Otherwise — procedurally — they are very similar.

(Source: Wikipedia)

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