rank
The rank of a matrix is how many of its columns are genuinely independent, that is, how many point in truly new directions once you throw away the ones that are just combinations of others. Surprisingly, you get the exact same number if you count independent rows instead, so a matrix has one single rank.
Put another way, rank is the dimension of the column space: how many real directions the matrix can actually reach when it acts on vectors. A 3-by-3 matrix written with three columns might look 3-dimensional, but if one column is the sum of the other two, its rank is only 2, and it secretly flattens space onto a plane.
Rank measures the true working power of a matrix. Full rank means it wastes nothing; lower rank means some directions have collapsed and information has been lost.
Both rows point the same way, so only one independent direction survives: rank 1.
Independent columns and independent rows always agree in count, which is why row reduction is a fair way to read off the rank.