Practice question · Multiple choice
The rank of a matrix can be defined by its rows or by its columns, and the two numbers are always equal, even for a 3 × 700 matrix. Why is that surprising?
Hints
- Ask which space each set of vectors lives in for a 3 × 700 matrix.
- Transposition swaps them, but does it obviously preserve the count of independent vectors?
Show the answer
D. Because the two counts are taken over completely different spaces
Why
Three vectors in ℝ⁷⁰⁰ and seven hundred in ℝ³, and the number of independent ones matches every time. It is a genuine theorem rather than a restatement, and it is what makes rank a property of the transformation rather than of how you chose to look at the array.
Practise Lines and Planes in Space
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