Sheet #6

Axiomatic definition. A (real) vector space V is a set with two operations called addition: $V\times V\to V$ and scalar multiplication: ${\mathbb R}\times V\to V$ that satisfy the following axioms for all $\mbox{\bf x}$, $\mbox{\bf y}$, $\mbox{\bf z}$ from V and all scalars ${\alpha}$, ${\beta}$ from ${\mathbb R}$.
\mbox{(2)}&\mbox{\bf x}+\mbox{\bf y}&=&\mb...
...ta})\mbox{\bf x}\\
\mbox{(10)}&1\cdot \mbox{\bf x}&=&\mbox{\bf x}

Definition. A subset W of a vector space V is called a subspace if W is itself a vector space under the addition and scalar multiplication defined on V

Example. Let $\mbox{\bf v}_1$, $\mbox{\bf v}_2$, ..., $\mbox{\bf v}_r$ be vectors in in a vector space V. Then the set $W = \{{\lambda}_1\mbox{\bf v}_1+{\lambda}_2\mbox{\bf v}_2+\dots+{\lambda}_r\mbox{\bf v}_r\}$ of all possible linear combinations of these vectors is a subspaces in V which is called the space spanned by $\mbox{\bf v}_1$, $\mbox{\bf v}_2$, ..., $\mbox{\bf v}_r$: $W=\mbox{Span}(\mbox{\bf v}_1, \mbox{\bf v}_2, \dots, \mbox{\bf v}_r)$.

Basis. Dimension is the number of vectors in a basis.

Vector spaces associated with a matrix. For a matrix $A\in{\cal M}(m,n)$ with m rows and n columns one can define the following four vector spaces

(1)         RA the subspace of ${\mathbb R}^n$ spanned by the rows of A (the row space of A)
(2)         CA the subspace of ${\mathbb R}^m$ spanned by the columns of A (the column space of A)
(3)         $\ker A$ $=\ \{\mbox{\bf x}\in{\mathbb R}^n : A\mbox{\bf x}=\mbox{\bf0}\}$ (the kernel (nullspace) of A)
(4)         $\mbox{im\ } A$ $=\ \{\mbox{\bf y}\in{\mathbb R}^m : A\mbox{\bf x}=\mbox{\bf y}$ for some $\mbox{\bf x}\in{\mathbb R}^n\}$ (the image (range) of A)

Relations: $C_A= \mbox{im\ } A$;          $\dim C_A = \dim R_A$;          $\dim(\ker A) + \dim(\mbox{im\ } A) = n$.

Definition. Rank of a matrix A is the dimension $\dim C_A = \dim R_A$.

Theorem. Let $\mbox{\bf x}_0$ be a particular solution of a system of linear equations $A\cdot \mbox{\bf x} = \mbox{\bf b}$ and let $\mbox{\bf v}_1$, $\mbox{\bf v}_2$, ..., $\mbox{\bf v}_r$ be a basis of the kernel of A. Then the general solution of the system has the form

\begin{displaymath}\mbox{\bf x} = \mbox{\bf x}_0+c_1\mbox{\bf v}_1+c_2\mbox{\bf v}_2+\dots+c_r\mbox{\bf v}_r\end{displaymath}