Question: Consider the system Ax = 0, where $$A=\begin{bmatrix} 1 & 6 & 2 & 5 &5\\ 1 & 2 & 2 & 1 & 3 \\ 2 &3&4&4&4\\ 2&-1&4&0&2 \end{bmatrix}$$
If $q$ is the number of free variables in the solution of this system and $p$ is the number of pivot columns in $A$ then $p + q = 5$
My answer: I do not know how I should solve this question/statement, I have tried to solve it for 20 min. Could someone please help me?
So I'm going to use rank-nullity. Every matrix can be thought of as a linear transformation from a vector space to another vector space. Here $A$ is a $4\times 5$ matrix and it take a vector in $\mathbb{R}^5$ to a vector in $\mathbb{R}^4$ by right multiplying a vector in $\mathbb{R}^5$ with $A$. The solution of the equation $Ax=0$ is null space of $A$, $null(A)=\{v\in \mathbb{R}^5|Av=0\}$. The image of $A$(also called column space of $A$), is defined as $Im(A)=\{v \in \mathbb{R}^4|v=Au,u\in \mathbb{R}^4\}$, are exactly the linear combinations of the colums of $A$. Rank-Nullity says that Given a linear transformation $T:V\rightarrow W$, $dim(V)=dim(null(A_T))+dim(Im(A_T))$, where $A_T$ is the matrix representation of $T$. In here, $V=\mathbb{R}^5$ and $W=\mathbb{R}^4$. So $dim(\mathbb{R}^5)=5=q+p$, as number of free variables is the dimension of the null space of $A_T$ and number of pivots is the dimension of column space of $A_T$.
The proof of Rank_nullity: Take a basis of $null(A)$,$\{v_1,v_2,....,v_k\}$, complete it to a basis for $V$,$\{v_1,...,v_k,w_1,...,w_m\}$. Let $v\in V, v=a_1v_1+a_2v_2+...+a_k v_k+b_1w_1+...+b_m w_m$, and $Av=a_1Av_1+...+a_k A v_k+b_1Aw_1+...+b_m A w_k=b_1Aw_1+...+b_m A w_m$. So we see that $B_C:=\{Aw_1,...,Aw_m\}$ is a spanning set for the column space of $A$. It remains to show $B_C$ is linearly independent. So take a linear combination of vectors in $B_c$, $0=a_1Aw_1+...a_mAw_m=A(a_1w_1+...+a_mw_m)$. Hence $ a_1w_1+...+a_mw_m \in null(A)$,so $a_1w_1+...+a_mw_m=b_1v_1+...+b_kv_k \Rightarrow a_1w_1+...+a_mw_m-b_1v_1-...-b_kw_k=0$ and by linearly independence of basis for $V$, all $a_i$ are $0$.