I have a school assignment which asks me to create a 49x49 matrix of rank 47 and none of the columns or rows are allowed to be parallel.
According to my textbook it says that the rank of a matrix is defined as the maximum amount of linearly independent columns of the matrix. Doesn't this mean that the kind of matrix they are asking me to create is impossible? Two columns that are parallel are also linearly dependent, right?
What am I misunderstanding here?
A matrix of size $n \times n$ and rank $m \le n$ has $(n-m)$ eigenvalues equal to zero. Then it means $(n-m)$ lines are a linear combination of the other $m$.
$$\mathbf{l}_{i} = \sum_{j=1}^{m} a_{ij} \cdot \mathbf{l}_{j} \ \ \ \ \ \forall \ i=m+1, \ \cdots, \ n$$
A specific case of linear combination is when one line is parallel to other line. So, there's a constant $a_{ij}$ such
$$\mathbf{l}_i = a_{ij} \cdot \mathbf{l}_{j}$$
To create a rank-deficient matrix with no parallel lines, just select two (or more) independent lines and do
$$\mathbf{l}_{i} = a_{ij} \cdot \mathbf{l}_{j} + a_{ik} \cdot \mathbf{l}_{k}$$
With $a_{ij} \ne 0$, $a_{ik} \ne 0$ and $j \ne k$