Row picture: rows of $A$ dot columns of $B$

Entry $(i,j)$ of $AB$ is row $i$ of $A$ dotted with column $j$ of $B$. Useful when interpreting rows as measurement functionals .

$(AB)_{ij} = \sum_k A_{ik}B_{kj}$ contracts on index $k$. Do not mix up the middle index: it contracts across the inner dimension. Inner dimensions must align because each dot pairs one row entry with one column entry along the same running index.

The row space of $AB$ is contained in the row space of $B$ when multiplying $AB$: rows of the product are combinations of rows of $B$. If $A$ is $1\times n$ and $B$ is $n\times p$, then $AB$ is $1\times p$.

Switch between column and row pictures depending on whether you track destinations or entrywise measurements. Both describe the same product .

Check your understanding. The tasks below rest on these ideas: Correct: the running index $k$ pairs a row entry of $A$ with a column entry of $B$ and is summed away. Not quite: $i$ and $j$ label the entry being computed, and the sum has $n$ terms, not one. Correct: each row of $AB$ mixes the rows of $B$ using the corresponding row of $A$ as weights, so the row space sits inside that of $B$. Not quite: it is built from $B$'s rows, need not fill the space, and is not orthogonal to them. Correct: the inner $n$ contracts, leaving the outer $1$ and $p$. Not quite: the others mismatch or transpose the surviving dimensions. Correct: the dot product underlying each entry pairs equal-length lists, forcing $A$'s row length to equal $B$'s column height. Not quite: it requires no equal determinants, squareness, or commuting.