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?
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.