031 Review Part 1, Problem 6

True or false: If  $A$  is a  $3\times 5$  matrix and  ${\text{dim Nul }}A=2,$  then  $A{\vec {x}}={\vec {b}}$  is consistent for all  ${\vec {b}}$  in  $\mathbb {R} ^{3}.$

Solution:
By the Rank Theorem, we have
${\begin{array}{rcl}\displaystyle {5}&=&\displaystyle {{\text{dim Col }}A+{\text{dim Nul }}A}\\&&\\&=&\displaystyle {{\text{dim Col }}A+2.}\end{array}}$
Hence,  ${\text{dim Col }}A=3.$
This tells us that  $A$  has three pivots.
Since  $A$  is a  $3\times 5$  matrix,
$A$  has a pivot in every row.
Therefore,  $A{\vec {x}}={\vec {b}}$  is consistent for all  ${\vec {b}}$  in  $\mathbb {R} ^{3}.$
So, the statement is true.

Final Answer:
TRUE

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