Linear Combinations and solutions [Columnwise Description of Matrices]

Question

Let $A$ be a $5\times 3$ matrix.

If $b = a_1 + a_2 = a_2 + a_3$ where $a_1, a_2, a_3$ are columns of $A$ then what can we conclude about the number of solutions of the linear system $Ax = b$?

Same Question for: when $b = a_1+a_2+a_3+a_4$

This query is from this Book page number 62

I am stuck at: let $a_1=[1,0,0]$, $a_2=[0,1,0]$ and $a_3=[0,0,1]$ but upon $a_1+a_2$ and $a_2+a_3$ these are not equal. such as $[1,1,0]$ and $[0,1,1]$

Answer 1

$A$ has $5$ rows. Let $\{e_1,e_2,e_3\}$ be the standard basis of $ \mathbb R^3.$

Then we have

$$Ae_j=a_j$$

for $j=1,2,3.$

Can you proceed ?

Answer 2

The rank of a matrix can be defined in many ways, in particular (i) maximum number of linearly independent rows (ii)maximum number of linearly independent columns (iii)maximum size of square sub-matrix with non-zero determinant. You should be comfortable with all 3 versions and know why they are equivalent. For your immediate purpose, (ii) is the most relevant. The rank of the augmented matrix of a system of linear equations is either the same as, or one more than, the rank of the coefficient matrix. The system is consistent iff the the rank of the augmented matrix is the same as the rank of the coefficient matrix. You can take it from there.