How to decompose a vector regarding complementary subspaces How to show that $U_1:=\langle(1,1,1)^T\rangle$ and $U_2:=\langle(1,0,0)^T,(0,0,1)^T\rangle$ are complementary subspaces and how to decompose $(1,2,3)^T$ regarding these ? Best Regards, Thomas
 A: You need to show that every $x \in \mathtt{R}^3$ (or whatever vector space you are interested in, your question doesn't specify one) can be uniquely represented as $$
  x = u_1 + u_2 \quad\text{where $u_1 \in U_1$ and $u_2 \in U_2$.}
$$
Since you already have a basis for $U_1$ and $U_2$, this is equivalent to showing that combining those basis vectors produces a basis for $\mathbb{R}^3$, which in turn is equivalent to showing that the matrix built from those basis vectors has determinant $\neq 0$. In your case, that means showing that $$
  \left|\begin{array}{lll}
    1 & 1 & 0 \\
    1 & 0 & 0 \\
    1 & 0 & 1
  \end{array}\right| \neq 0 \text{.}
$$
To decompose a vector into $u_1, u_2$, one way is to find it's coordinates relative to the basis created from the basis vectors of the subspaces. You can then easily combine one vector per subspace, whose sum will produce the original vector.
A: The best way to do this is to solve the system of equations $Ax=b$, where
$$A=\left[\begin{array}{ccc}1&1&0\\1&0&0\\1&0&1\end{array}\right]\quad\text{and}\quad b= \left[\begin{array}{c} 1\\2\\3\end{array}\right].$$
With the numbers you have, there are shortcuts, as others have suggested.
A: Notice that $U_2$ has zero in the $y$ coordinate, so whatever part is due to $U_1$ must account for the 2 in your vector. Thus $(1,2,3) = (2,2,2)+(-1,0,1)$ is the place to start.
Edit: Whoops! For the first part, the easiest thing is to write the standard basis $(1,0,0) ,(0,1,0),(0,0,1)$ in terms of your vectors since they're so simple.
