I found on Stat Trek an example of a matrix that has a rank of 1. As far as I know
Here is the matrix:
[1 2 3]
[2 4 6]
Linearly Dependent:
$x_d=\sum_{i=1}^{d-1}a_ix_i,a_i\in \mathbb{R}$ where $x_i$ is a vector
Column or Row Rank:
Size of the largest subset of columns or rows that are linearly independent
The answer was the above matrix has a rank of 1, because row 1 times 2 turns into row 2, but then they state "Matrix A has only one linearly independent row, so its rank is 1", but this doesn't make sense cause you can just times row 2 by $\frac{1}{2}$ to get row one since $a_i$ just has to be a real number, therefore it's not linearly independent.
Yes, we all agree the row rank is 1. so row 1 and row 2 are linearly dependent.
No. By definition row 1 and row 2 are NOT independent as a set.
Individually, each row is independent.
If this makes you confused. I have a real life analogy here.
Consider John and Lisa who got married on January 1, 2021. They now depend on each other.
Lisa needs John to work full time to make a living for the entire family.
John, on the other hand needs someone to keep their home nice and clean. He also needs to have a delicious meal as well as great conversation after the meal.
But prior to 2021, they are independent. Each comes from a different family. Each has his or her own budget, hobby and interest. Each lives in his or her small world.
Now can you argue John and Lisa are dependent as a team( which has rank 1) does NOT make sense? No, it makes perfect sense. It depends your perspective:
As individuals, they are independent. As a member of new formed family, they are indeed dependent!
There is NO contradiction and No equations either!