Multiplication of matrices properties

[bonfire09]

Homework Statement

Let e_j denote the jth unit column that contains a 1 in the jth
position and zeros everywhere else. For a general matrix A_n×n, describe the following products. (a) Ae_j (c) e^T_iAe_j?

Homework Equations

Rows and Columns of a Product
Suppose that A = [a_ij] is m × p and B = [b_ij] is p × n.
• [AB]_i∗ = A_i∗B [( ith row of AB)=( ith row of A) ×B]. (3.5.4)

• [AB]_∗j = AB_∗j [ (jth col of AB)=A× ( jth col of B)]. (3.5.5)

• [AB]_i∗ = a_i1B_1∗ + a_i2B_2∗ + · · · +a_ipB_p∗ =Ʃ a_ikB_k∗. (3.5.6)

• [AB]_∗j = A_∗1b_1j + A_∗2b_2j + · · · + A_∗pb_pj=Ʃ A_∗kb_kj (3.5.7)

These last two equations show that rows of AB are combinations of
rows of B, while columns of AB are combinations of columns of A.

The Attempt at a Solution

For parts a and c I am not even sure what they are even asking for. When its saying e_j is a unit column does that mean like this (1 0 0...0) as an example? For part A wouldn't the solution of Aej just just be a linear combination a column of A and the entries of ej as scalars?

 

[haruspex]

Yes, it means that e_j = (δ_ij)^T, i.e. 1 where i=j and zero elsewhere.
From your matrix multiplication formulae, can you deduce a formula for B being a column vector instead of a matrix?