Let W1=span{(1,0,0)}W_1 = \text{span}\{(1, 0, 0)\}W1=span{(1,0,0)} and W2=span{(0,1,0),(0,0,1)}W_2 = \text{span}\{(0, 1, 0), (0, 0, 1)\}W2=span{(0,1,0),(0,0,1)} in R3\mathbb{R}^3R3. Is R3=W1⊕W2\mathbb{R}^3 = W_1 \oplus W_2R3=W1⊕W2 (direct sum)?
Yes, because W1∩W2={0}W_1 \cap W_2 = \{\mathbf{0}\}W1∩W2={0} and dim(W1)+dim(W2)=3\dim(W_1) + \dim(W_2) = 3dim(W1)+dim(W2)=3
No, because W1W_1W1 and W2W_2W2 overlap
No, because W1∪W2≠R3W_1 \cup W_2 \neq \mathbb{R}^3W1∪W2=R3
Yes, but only if we use orthogonal complements