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- TL;DR Summary
- Consider ##U=\{\textbf{x} \in \mathbf{R}^3; x_1+x_2+3x_3=0\}## and ##U=\{t(1,1,2)\in\mathbf{R}^3; t \in \mathbf{R}\}##. Show that ##\mathbf{R}^3=U \bigoplus V##.

If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in \mathbf{R}^3## can be expressed as a sum of ##\textbf{u}=(x_1,x_2,x_3) \in U## and ##\textbf{v}=(t,t,2t) \in V## for ##t \in \mathbf{R}##?