Question

Show that W={(x,-3x,2x)|x€R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies R³ is equal to direct sum of W and U.

Answer 1

$(0,0,0)\isin W$

W is closed under vector addition:

$(x_1,-3x_1,2x_)+(x_2,-3x_2,2x_2)=(x_1+x_2,-3x_1-3x_2,2x_1+2x_2)\isin R^3$

W closed under scalar multiplication:

$a(x,-3x,2x)=(ax,-3ax,2ax)\isin R^3$

So, W is a subspace of R³.

The sum of subspaces U and W is direct if and only if every vector x∈U+W can be represented uniquely as x=u+w where u∈U and w∈W.

So, a basis for subspace U of R³:

$\{(1,0,0),(0,1,0),(0,0,1)\}$