Another method? - Matrix & Linear Independence

[RajdeepSingh7]

Question :
Let A be a 7 × 4 matrix. Show that the set of rows of A is linearly dependent.


Answer:
The row vectors of a matrix are linearly independent if and only if the rank of the matrix is equal to the number of rows in the matrix.
Since rank (A) = 4 , and the number of rows in the matrix is 7, the row vectors are linearly dependent.



I am sure that my answer is right, but I was considering or wondering if there was an alternative method, because the actual answer is worth 4 marks, so was wondering if there was a more mathematical sound proof possible to prove the answer?

 

[HallsofIvy]

The rows of a 7 by 4 matrix are members of R⁴ which has dimension 4- any set of more than 4 vectors must be dependent.

Your basic statement that a 7 by 4 matrix has rank 4 is NOT correct. The matrix
$$\begin{bmatrix}1& 1 & 1 & 1\\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \end{bmatrix}$$
does not have rank 4.