Module 2 : Fundamentals of Vector Spaces
Section 2 : Vector Spaces
 
Example 21 Basis, Span and Sub-spaces

  1. Two dimensional plane passing through origin of $R^{3}$. For example, consider the set $S$ of collection of all vectors MATH where $\alpha,\beta\in R$ are arbitrary scalars and MATH i.e. MATHThis set defines a plane passing through origin in $R^{3}.$ Note that a plane which does not pass through the origin is not a sub-space. The origin must be included in the set for it to qualify as a sub-space.

  2. Let $S=\{\QTR{bf}{v}\}$ where MATH and let us define span of $S$ as MATH where $\alpha\in R$ represents a scalar. Here, $\left[ S\right] $ is one dimensional vector space and subspace of $R^{5}$

  3. Let MATH where MATH                                        ----(5) Here span of $S$ (i.e. $[S]$) is two dimensional subspace of $R^{5}$.

  4. Consider set of $n^{th}$ order polynomials on interval $[0,1]$. A possible basis for this space is
    MATH                      ----(6)
    Any vector $p(t)$ from this space can be expressed as
    MATH                                ---(7)
    Note that $[S]$ in this case is $(n+1)$ dimensional subspace of $C[a,b]$.

  5. Consider set of continuous functions over interval, i.e. $C[-\pi,\pi].$ A well known basis for this space is
MATH ---(8)
---(9)

It can be shown that $C[-\pi,\pi]$ is an infinite dimensional vector space.

 

    6.  The set of all symmetric real valued $n\times n$ matrices is a subspace of the set of all real valued           $n\times n$ matrices. This follows from the fact that matrix MATH is a real values symmetric           matrix for arbitrary scalars $\alpha,\beta\in R$ when MATH and MATH