Row and column vector spaces

The linear span $\langle U \cup V\rangle$ is called the sum of the subspaces $U$ and $V$:

If $U \cap V=0$, then the sum $U+V$ is said to be direct, and one writes $U \oplus V$.

Let $V=V_{1} \oplus V_{2}$ and $X=X_{1}+X_{2}=X_{1}^{\prime }+X_{2}^{\prime }$ be two expressions of a vector $X \in V$ as a linear combination of vectors $X_{1}, X_{1}^{\prime } \in V_{1}$ and $X_{2}, X_{2}^{\prime } \in V_{2}$. Then we have $X_{1}-X_{1}^{\prime }=X_{2}^{\prime }-X_{2} \in V_{1} \cap V_{2}$, and since $V_{1} \cap V_{2}=0$, it follows that $X_{1}=X_{1}^{\prime }, X_{2}=X_{2}^{\prime }$.

Prove the converse: if the expression $X=X_{1}+X_{2}, X_{i} \in V_{i}, i=1,2$, is unique for every vector $X \in V$, then the sum $V=V_{1}+V_{2}$ is direct. More generally: the sum $V$ of subspaces $V_{1}, \ldots , V_{k} \subset \mathbb {R}^{n}$ is called a direct sum $V=V_{1} \oplus \ldots \oplus V_{k}$ if every vector $X \in V$ has a unique expression of the form $X=X_{1}+\ldots +X_{k}$ with $X_{i} \in V_{i}$.

Let $V, V_{1}$ and $V_{2}$ be linear spans in $\mathbb {R}^{n}$, with $V \subset V_{1}+V_{2}$. Is it always true that $V=V \cap V_{1}+V \cap V_{2}$? What can be said about this relation in the special case $V_{1} \subset V$?

Let $V$ be a linear span in $\mathbb {R}^{n}$. If $V=U \oplus W$ is a decomposition into a direct sum, then the span $W$ is called a complement of $U$, and $U$ a complement of $W$, in $V$. Is the complement of $U$ in $V$ uniquely determined? Compare $W$ with the set-theoretic notion of the complement $V / U$ (see § 5, Ch. 1).

Show that the vectors $X_{1}=(1,2,3), X_{2}=(3,2,1)$ are linearly independent; consider the linear span $V \left\langle X_{1}, X_{2}\right\rangle$; show that the vector $X=(-5,2,9)$ is contained in $V$, and find its coordinates in the basis $X_{1}, X_{2}$; find in $\mathbb {R}^{3}$ at least one complement of $V$.

Show that a system of vectors $X_{1}, \ldots , X_{n}$ from $\mathbb {R}^{n}$ spans $\mathbb {R}^{n}$ if and only if it is linearly independent.

Show that any linearly independent system of vectors $X_{1}, \ldots , X_{k}$ from a linear span $V \subset \mathbb {R}^{n}$ can be embedded in some basis system for $V$.

Let $U$ and $V$ be linear spans in $\mathbb {R}^{n}$. Prove that if $U \cap V=0$, then $\operatorname {dim}(U+V)=\operatorname {dim} U+\operatorname {dim} V$.

Find the rank of the system of vectors $(0,1,1),(1,0,1),(1,1,0)$.