We can view the matrix product $AB=C$ in (at least) four different ways:

1. Each entry of $C$ is a dot product: The entry $C_{i,j}=(\text{row }i\text{ of }A)\cdot (\text{column }j\text{ of }B)=\sum _{k=1}^n{A}_{i,k}{B}_{k,j}.$
2. Each column of $C$ is $A$ times the corresponding column of $B$: $C_{\cdot ,j}=A{B}_{\cdot ,j},$ so $$C_{\cdot ,j}=B_{1,j}{A}_{\cdot ,1}+\cdots +B_{n,j}{A}_{\cdot ,n}$$ In other words, the columns of $C$ are linear combinations of the columns of $A.$
3. Each row of $C$ is the corresponding row of $A$ multiplied by $B:$ $C_{i,\cdot }=A_{i,\cdot }B,$ so $$C_{i,\cdot }=A_{i,1}{B}_{1,\cdot }+\cdots +A_{i,n}{B}_{n,\cdot }.$$ In other words, the rows of $C$ are linear combinations of the rows of $B.$
4. $C$ is a sum of column $\times$ row outer products: $$C=AB=a_1{b}_1^{\mathrm{\top }}+a_s{b}_2^{\mathrm{\top }}+\cdots +a_n{b}_n^{\mathrm{\top }}.$$