What is a geometric interpretation of all these information?

[mathmari]

Hey!

We have the tableau $\begin{pmatrix}
\left.\begin{matrix}
1 & 0 & \alpha \\
0 & 1 & \beta \\
0 & 0 & 0
\end{matrix}\right|\begin{matrix}
c\\
d\\
0
\end{matrix}
\end{pmatrix}$

Since there is a zero-row, we conclude that the column vectors are linearly dependent.

The number of linearly independent row- and column vectors is the same. And from the tableau we get that there are $2$ linearly independent row- and column vectors. Therefore the dimension of the the vector space spanned by the solumn vectors is $2$. And this is also equal to the rank of the matrix.

The dimension of the solution space is equal to the numer of free variables, so $1$, right?

Is everything correct so far? (Wondering)

What is a geometric interpretation of all these information? Do we get that two column vectors are either a multiple of each other or they are on the same line? Or is there also an other interpretation? (Wondering)

 

[jvicens]

Hello!

Yes, your conclusions are correct so far. The fact that there is a zero-row in the tableau indicates that the column vectors are linearly dependent, meaning that one column vector can be written as a linear combination of the other two. This also means that the rank of the matrix is 2, since there are only two linearly independent column vectors.

As for the dimension of the solution space, you are correct that it is equal to the number of free variables, which in this case is 1. This means that there is only one parameter needed to describe all possible solutions to the system of equations represented by the tableau.

In terms of geometric interpretation, you are also correct that the two column vectors are either a multiple of each other or they lie on the same line. This can be seen by looking at the third column, which is all zeros, indicating that the third variable is a free variable. This means that the solutions lie on a line, or one-dimensional subspace, in three-dimensional space.

I hope this helps clarify things for you! Let me know if you have any other questions.