Multiplying matrix units and standard basis vectors

[bekkilyn]

Hello all, I don't have a question on homework specifically, but I need clarification on something I'm reading in the textbook.

I will be starting an abstract algebra class in the spring and it's been quite a few years since I've had linear algebra, so I'll be reviewing that material before the abstract algebra class starts. I've also started the first chapter of the book, Algebra, by Michael Artin. So far, I've been able to make sense of most of what I've read, but I'm stuck on this one formula at the top of page 10.

He states that the formulas for multiplying matrix units and standard basis vectors are:

e_ije_j = e_i

and

e_ije_k = 0 if j ≠ k

I understand from the previous page that the matrix unit e_ij has a 1 in the ij position as its only non-zero entry and based on an example from the previous page, you can show a standard m x n matrix as a linear combination that includes e_ij.

My confusion is in figuring out what the e_i and e_j are in the above formula. The bottom of the previous page discusses a column vector e_i but I wasn't sure how to connect this vector to the above formula or how I should multiply it with e_ij to get e_j.

Maybe if I saw a couple of concrete examples of what e_i and e_j are as compared with e_ij, it would help me clear up this confusion.

Thanks for any help on this question!

 

[jbunniii]

$e_{i}$ is a column vector with 1 in the $i$th row, and zeros everywhere else.

Simple example with a 4x4 matrix and 4x1 vectors:

$$\left(\begin{array}{cccc}0&0&0&0\\
0&0&1&0\\
0&0&0&0\\
0&0&0&0\end{array}\right)
\left(\begin{array}{c}0\\0\\1\\0\end{array}\right) =
\left(\begin{array}{c}0\\1\\0\\0\end{array}\right)$$
which is the same as $e_{23} e_3 = e_2$

 

[D H]

My confusion is in figuring out what the e_i and e_j are in the above formula. The bottom of the previous page discusses a column vector e_i but I wasn't sure how to connect this vector to the above formula or how I should multiply it with e_ij to get e_j.

They're vectors. In particular, they are column vectors. Right up front, Artin defines vectors as being either equivalent to a 1xn or an nx1 matrix. Think of e_j as meaning e_j1, with the column index of 1 implied by the fact that this N vector is equivalent to an Nx1 matrix.

 

[bekkilyn]

Thank you both! The matrix example and the additional column vector description was very helpful as I can now picture what's going on and how the matrix and vectors are indexed in reference to each other. Much clearer now!