Defining term with $3$-vectors and $3 \times 3$ matrices I don't normally ask questions, but my professors isn't responding back to me. I do not want an answer or anything, just want a point in the right way so I can figure this set of question out. 
The question is:
 $3$-vectors with Cartesian coordinates and $3 \times 3$ matrices: 
 Scalar multiplication (scalar times matrix)
 Matrix multiplication (matrix times matrix)
At the moment, I think I have the following answer as 
$s[M]$ which is $s \cdot M_{ij}$ for the matrix
$s[V]$ which is $s\cdot V_i$ for the Cartesian in a vector form. 
However I'm not so sure if that what the question want. Does it want the $VM$, where $V$ is the  scalar in question? 
Sorry if this is confusing. To be honest,  I'm very confused about the problem too. 

Let $M$ be a $3 \times 3$ matrix given by columns $M_1, M_2, M_3$ (so each of $M_1, M_2, M_3$ is a three vector in column format). Then we have $$M = [M_1|M_2|M_3].$$
Let Let $W$ be a $3 \times 3$ matrix given by rows $W_1, W_2, W_3$ (so each of $W_1, W_2, W_3$ is a three vector in row format). Then we have $$\pmatrix{W_1\\W_2\\W_3}$$ 
Let $$V = \pmatrix{v_1\\v_2\\v_3}$$ be a 3-vector in column format. Let $Y$ also be a vector, this time in row form: $$Y = \pmatrix{y_1&y_2&y_3}.$$
Let $s$ be a scalar. Then, as you said, $s[V] = s\cdot V = \pmatrix{sv_1\\sv_2\\sv_3}$
Also, $$s[M] = s\cdot M = [s\cdot M_1|s\cdot M_2|s\cdot M_3],$$ where multiplication of a vector by a scalar is defined above.
Multiplication of two vectors $$Y\cdot V = y_1\cdot v_1 + y_2\cdot v_2 + y_3\cdot v_3.$$
Finally, we have matrix multiplication:
$$W\cdot M =\pmatrix{W_1\cdot M_1|W_2\cdot M_2|W_3\cdot M_3},$$ where the vectors $W_i$ and $M_i$ are multiplied as described above in the multiplication of $V$ and $Y$. 
