Find examples of subspaces $W_1$ and $W_2$ of $R^3$ such that $dim(w_1)$ >$dim(w_2)$ >$0$ and
a). $dim(w_1 \cap w_2)$ = $dim(w_2)$
b). $dim(w_1+w_2)$=$dim(w_1)$+$dim(w_2)$
c). $dim(w_1+w_2)$< $dim(w_1)+dim(w_2)$
For a). I have $w_1$= {$(1,0,0),(0,1,0),(0,0,1)$}, $w_2$={$(1,0,0),(0,1,0),(1,1,0)$}
For b). $w_1$={$(1,2,3),(2,3,4),(3,4,5)$}, $w_2$={$(4,5,6),(6,7,8)$}
For c). $w_1$={$(1,0,0),(0,1,0),(0,0,1)$}, $w_2$={$(1,0,0),(1,2,3)$}
Are these correct?
Best Answer
I assume that the notation $\{v_1,v_2\}$ actually means the span of the given vectors.
a) is correct, however we don't need to mention the vector $(1,1,0)$ in $w_2$, as it's just the sum of the other two vectors.
b) is incorrect, both your $w_1$ and $w_2$ equal to $\mathrm{span}\{(1,2,3),\ (1,1,1)\}$.
c) is correct.