Could intersection of a subspace with its complement be non empty. If that is possible could you please correct my understanding about complement of a subspace. 
From what i recall from set theory. A complement of a set B is the set U - B where U is the universal set. So B and U - B are disjoint, but in case of subspaces of a finite dimensional vector space all subspaces of the same dimension have origin in them, hence the intersection of such a subspace with it's complement is n't disjoint. Is n't this contradictory ? 
Any help would be much appreciated.
 A: The word "complement" has different meanings depending on the context. In Set Theory, we often talk about "[relative] complement", which is the concept you describe. As you note, a set and its (relative) complement are disjoint. Moreover, the relative complement is unique: there is one and only one set that qualifies as the relative complement of $A$.
In linear algebra, though, "complement of a subspace" has a different meaning. We say that a subspace $\mathbf{Z}$ of the vector space $\mathbf{V}$ is a complement of the subspace $\mathbf{W}$ if and only if (i) $\mathbf{V}=\mathbf{W}+\mathbf{Z}$; and (ii) $\mathbf{W}\cap\mathbf{Z}=\{\mathbf{0}\}$. In general, there are many different possible complements, and none are disjoint from $\mathbf{W}$ (however, since $\{\mathbf{0}\}$ is the smallest that the intersection of two subspaces can be, we usually say the intersection is "trivial"). For example, if $\mathbf{V}=\mathbb{R}^2$ (as a real vector space) and $\mathbf{W}$ is the $x$-axis, then any line through the origin except the $x$-axis is a complement of the $x$-axis.
There is even a further concept of "complement" in linear algebra, when we have a notion of "inner product"; then we talk about the orthogonal complement of a set/subspace: if $\mathbf{V}$ is a vector space with inner product $\langle \cdot,\cdot\rangle$, then the orthogonal complement of a set $S$ is the set of all vectors that are orthogonal to $S$, and is denoted $S^{\perp}$; in general, $S\cap S^{\perp}\subseteq \{\mathbf{0}\}$.
So, in summary: the same word ("complement") has different meanings depending on context; you are trying to apply one meaning (set-theoretic) in the wrong context (linear algebra). You should look for the linear meaning instead. If you think it's bad to have the same word mean different things... well, you are right, but since the contexts are so different, it usually doesn't matter. (If you think that's bad, just wait until you run into the term "normal"...)
