## Outer Product

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

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.

$\dpi{300} \bg_white a_1 \! \cdot \! b_1 \ + \ a_2 \! \cdot \! b_2 \ + \ a_3 \! \cdot \! b_3 \ + \ \ldots$

$\dpi{300} \bg_white \sum^n_{i=1} a_i \cdot b_i$

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.

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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