Linear algebra 3

Question

Given the set of all vectors in R^3 such that ||U||∞ < 10 , does this form a subspace of R^3 ?

Answer 1

Can you multiply a vector with norm less than 10 by some scalar and get one with norm greater than or equal to 10? Or, can you add two vectors with norm less than 10 and get a vector with norm greater than or equal to 10? Either way would show it not to be a subspace of R^3. 

 

The infinity norm of (a_1,a_2,a_3) is defined to be max{|a_1|,|a_2|,|a_3|}. Then clearly yes in the first case: take (0,0,5) and multiply by 2 to get something with norm greater than or equal to 10, hence not a subspace. Or, add (0,0,5) to (0,0,7) to get (0,0,12).

 

Visually, we have the intersection of the loci {(x,y,z) | -10 < x < 10},

{(x,y,z) | -10 < y < 10}, and

{(x,y,z) | -10 < z < 10}

hence the interior of a solid cube with side lengths 20, centered at the origin.