031 Review Part 1, Problem 8

True or false: Let  $W$  be a subspace of  $\mathbb {R} ^{4}$  and  ${\vec {v}}$  be a vector in  $\mathbb {R} ^{4}.$  If  ${\vec {v}}\in W$  and  ${\vec {v}}\in W^{\perp },$  then  ${\vec {v}}={\vec {0}}.$

Solution:
Since  ${\vec {v}}\in W^{\perp },$  we know  ${\vec {v}}$  is orthogonal to every vector in  $W.$
In particular, since  ${\vec {v}}\in W,$  we have that  ${\vec {v}}$  is orthogonal to  ${\vec {v}}.$
Hence,
${\vec {v}}\cdot {\vec {v}}=0.$
But, this tells us that  ${\vec {v}}={\vec {0}}.$
Therefore, the statement is true.

Final Answer:
TRUE

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