Seeing Rows Combine

Looking beyond the obvious relationships between columns or rows in a matrix and finding the less-obvious connections that we know are there.

The matrices that you encounter in Linear Algebra textbooks often feel "designed" to have column relationships that are very easy to spot, usually for teaching purposes.

The exercise of finding these obvious column relationships is to demonstrate the idea that a matrix can be broken down into simpler matrices, and that the columns of a matrix can be seen as a combination of other columns.

If the column relationships are made to be obvious, the row relationships are usually more subtle (and vice versa). But they are always there, and they can be seen if you squint hard enough!

Take this one, for example.

$\left[\begin{array}{ccc}2& 4& 7\\ 3& 6& 4\\ 4& 8& 9\end{array}\right]$

You don't have to stare at the grid too long to see that the second column is the first column multiplied by 2.

(The third column is whatever; no connection is easily perceptible).

Convert that 2x column multiplier into a variable, for example $m$, and you can see the connection between columns one and two even more clearly.

$\left[\begin{array}{ccc}a& ma& x\\ b& mb& y\\ c& mc& z\end{array}\right]$

It's now quite clear, and in fact it is literally spelled out, that the second column is the first column multiplied by $m$. The third column is still whatever, which we can express by using totally unrelated, independent variables.

In the language of "independence" between columns, we say that the first and third columns are independent, and the second column is dependent on the first column, because it is a simple multiple of that column.

In technical terms, the rank of this matrix is 2, because out of three columns, only two are independent, while the third is a linear combination of the first two (or in this simple case, it's just a multiple of the first column).

This observation implies that we can start with a simpler 2-column matrix, and combine those 2 columns in certain ways to get back to the original 3-column matrix.

The trick, as it is taught in classrooms, is to get rid of all of the dependent columns first.

$\left[\begin{array}{cc}a& x\\ b& y\\ c& z\end{array}\right]$

Now, how do we get from this 2-column matrix to our original 3-column matrix?

We can derive a second matrix, with that factor $m$ inside it, to specify a combination of the columns from the simpler two-column matrix of $\mathrm{(a,\; b,\; c)}$ and $\mathrm{(x,\; y,\; z)}$. These combinations will form the original $\mathrm{3x3}$.

Here are the three coefficients of linear combination for our 2-column matrix, expressed as a $\mathrm{2x3}$ matrix itself.

$\left[\begin{array}{ccc}1& m& 0\\ 0& 0& 1\end{array}\right]$

We now have two matrices to work with.

The matrix with columns $\mathrm{(a,\; b,\; c)}$ and $\mathrm{(x,\; y,\; z)}$ contains the basic ingredients of the $\mathrm{3x3}$, and this new $\mathrm{2x3}$ matrix is like a recipe, specifying how to combine the columns of our ingredients matrix to get the $\mathrm{3x3}$ matrix from the beginning.

Now let's turn the tables and shift our perspective from thinking about combining columns to combining rows instead.

Let's convert the "recipe" matrix into the "ingredients" matrix now.

The columns $\mathrm{(a,\; b,\; c)}$ and $\mathrm{(x,\; y,\; z)}$ of the simpler 2-column matrix is now a recipe for combining the rows $\mathrm{[1,\; m,\; 0]}$ and $\mathrm{[0,\; 0,\; 1]}$ to get the exact same $\mathrm{3x3}$ matrix as before.

The conceptual trick is to convert those two independent columns $\mathrm{(a,\; b,\; c)}$ and $\mathrm{(x,\; y,\; z)}$ into three rows: $\mathrm{[a,\; x]}$, $\mathrm{[b,\; y]}$, and $\mathrm{[c,\; z]}$, stacked vertically. Same matrix, different orientation.

Each of these new two-item rows now provide the coefficients we need to scale each of the rows $\mathrm{[1,\; m,\; 0]}$ and $\mathrm{[0,\; 0,\; 1]}$ and combine them back to our original $\mathrm{3x3}$ matrix.

Since we have three recipe rows, we have three combinations of $\mathrm{[1,\; m,\; 0]}$ and $\mathrm{[0,\; 0,\; 1]}$ to do. The operations line up perfectly.

We scale $\mathrm{[1,\; m,\; 0]}$ by $a$ and $\mathrm{[0,\; 0,\; 1]}$ by $x$, and so on.

The three rows of our final $\mathrm{3x3}$ matrix consist of three linear combinations of $\mathrm{[1,\; m,\; 0]}$ and $\mathrm{[0,\; 0,\; 1]}$ as follows.

$a\mathrm{[1,\; m,\; 0]}+x\mathrm{[0,\; 0,\; 1]}=\mathrm{[a,\; ma,\; x]}$

$b\mathrm{[1,\; m,\; 0]}+y\mathrm{[0,\; 0,\; 1]}=\mathrm{[b,\; mb,\; y]}$

$c\mathrm{[1,\; m,\; 0]}+z\mathrm{[0,\; 0,\; 1]}=\mathrm{[c,\; mc,\; z]}$

These three operations return us back to our original matrix, this time built as a combination of rows, not columns.

$\left[\begin{array}{ccc}a& ma& x\\ b& mb& y\\ c& mc& z\end{array}\right]$

Same end-result matrix, but with different "ingredients" and a different "recipe."

Let's now return all the way back to our very first example, the one without any variables.

Here we had a $\mathrm{3x3}$ matrix, and the column derivation was so obvious, it practically forced us to see the matrix as a collection of columns.

$\left[\begin{array}{ccc}2& 4& 7\\ 3& 6& 4\\ 4& 8& 9\end{array}\right]$

Now, let's squint. Can we see past the column ingredients, and percieve the same grid as a combination of rows?

What two row ingredients could possibly combine to give us this matrix?

In practice, it's quite a stretch to look at the row $\mathrm{[2,\; 4,\; 7]}$ in the example and say, "Oh! That first row $\mathrm{[2,\; 4,\; 7]}$ is actually a linear combination of two row vectors $\mathrm{[1,\; 2,\; 0]}$ and $\mathrm{[0,\; 0,\; 1]}$, where the first row is scaled by 2, and the second row is scaled by 7..."

But it's true. If we can combine a couple of columns into a larger matrix, we can always think of the same matrix as a combination of rows instead.

What seems most magical to me is that the process for revealing the hidden row "ingredients" is to simply write down the "recipe" matrix for the column ingredients. The recipe for the column ingredients is identical to the ingredients for the row recipe!

The connection is almost never easy to see both ways without deliberately pulling the matrix apart, but it is always there, waiting to be revealed.