What is the difference between the answers of the following 2 questions? The 2 questions are given below:



I already know an answer to this question using proof by contradiction.



And Also, I know a direct way of proving this question.
Can I say that the proof of question 619 follows directly from question 178?
EDIT:
I see that problem 619 preserves linear independence, but how it preserves that this is the maximum linearly independent set? Could anyone help me in this please?
 A: You can "almost" apply the result of 178 to 619, as long as you can translate a statement about linear independence to a question about rank.
Lemma : Let $F \subset K$ be fields and $n > 0$ a positive integer. Let $S$ be a set of vectors in $F^n$. Let $V$ be the vector space spanned by $S$ over $F$. Now, consider $S$ as a subset of $K^n$, and let $W$ be the subspace spanned by $S$ over $K$. Then, $\dim_F V = \dim_K W$ i.e. the "rank" of $S$ doesn't change upon going to a larger subspace.
Proof : Note that $\dim_F V$ is the size of the largest $F$-linearly independent subset of $S$. Similarly, $\dim_K W$  is the size of the largest $K$-linearly independent subset of $S$.
Therefore, let $S' \subset S$ be any set of vectors.
Claim : $S'$ is $F$-linearly independent if and only if $S'$ is $K$-linearly independent.
One side of the proof ($K$-lin.ind implies $F$-lin ind.) is obvious from $F \subset K$. The other side is exercise $178$.
Therefore, it is clear, that the size of the maximal $F$-lin.ind subset of $S$ is equal to the size of the maximal $K$-lin.ind subset of $S$. Which shows the desired equality.

Once this is done, if $A$ is a rank $r$ matrix over $F$, then its row rank is $r$, so taking its row space as the set $S$ above, the "rank" doesn't change viewing $S$ as the row space of $A$ over $K$. Thus rank is preserved. 
