# What do the columns of a rectangular matrix mean?

I have the following system of equations

$$x_1+x_2+x_3=1$$

$$x_4+x_5+x_6=1$$

from a linear programming example in standard form. Putting it into a matrix form ($Ax=b$)

\begin{equation*} \begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1\\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6 \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix} \end{equation*}

Do the columns in the $A$ matrix, represent vectors in a 2D space?

I think of columns of the $A$ matrix as vectors and the terms in the $x$ vector as 'weighting' elements such that the $b$ vector is a weighted sum of the columns of the $A$ matrix.

For example

\begin{equation*} \begin{pmatrix} 1 & 2 & 3\\ 5 & 7 & 8 \\ 3 & 1 & 12\\ \end{pmatrix} \begin{pmatrix} x\\ y \\ z\\ \end{pmatrix} = \begin{pmatrix} 1x+2y+3z\\ 5x+7y+8z \\ 3x+1y+12z\\ \end{pmatrix} = x\begin{pmatrix} 1\\ 5 \\ 3\\ \end{pmatrix} + y\begin{pmatrix} 2\\ 7 \\ 1\\ \end{pmatrix} + z\begin{pmatrix} 3\\ 8 \\ 12\\ \end{pmatrix} \end{equation*}

Is this the same for the first matrix equation I mentioned?

I'm confused because the original set of equations has $6$ variables while if my understanding of the columns of the $A$ matrix is correct, we have $6$ 2D vectors. Furthermore, $3$ of the columns of the $A$ matrix are identical, so the rank is only $2$. The 2 equations at the top seem to represent planes in a 3D space while the column vectors of the $A$ matrix represent vectors on a 2D plane.

Any help in sorting all of this out in my brain is much appreciated!

• What is the question? – polfosol Oct 3 '16 at 17:37
• You're absolutely right with all you said except that the two equations represent a hyperplane in 6D each. – Michael Hoppe Oct 3 '16 at 19:28
• @MichaelHoppe I see, it's good to know I don't have everything upside down! I guess this is not completely crucial for my understanding, but what are two equations in the 6D space? Is it that they're more multidimensional regions rather than planes? It seems like they should be regions because (for example in equation 1), the region should extend to infinity the x4, x5 and x6 directions. – Urmi Oct 5 '16 at 4:50
• As you have six unknowns, the first equation reads $x_1+x_2+x_3+0x_4+0x_5+0x_6=0$. – Michael Hoppe Oct 5 '16 at 8:23
• Gilbert Strang uses the terms "row picture" (intersection of hyperplanes) and "column picture" (linear combination of vectors). It looks as though you are used to the "row picture" and are perplexed by the "column picture". As soon as monomials of degree greater than $1$ are introduced, the "row picture" acquires an algebro-geometric flavor, and the "column picture" can no longer be used. – Rodrigo de Azevedo Oct 6 '16 at 19:54