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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}\in W}   and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}\in W^{\perp },}   then  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}={\vec {0}}.}

Solution:
Since  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}\in W^{\perp },}   we know  ${\vec {v}}$  is orthogonal to every vector in  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W.}
In particular, since  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}\in W,}   we have that  ${\vec {v}}$  is orthogonal to  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}.}
Hence,
${\vec {v}}\cdot {\vec {v}}=0.$
But, this tells us that  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}={\vec {0}}.}
Therefore, the statement is true.

Final Answer:
TRUE

Return to Review Problems