Geometry of linear equations: linear spaces and subspaces

Jul 18

Geometry of linear equations: linear spaces and subspaces

Geometry of linear equations: linear spaces and subspaces

$R^n$ is a linear space in the following sense: for any $x,y\in R^n$ and any $a,b\in R$ the linear combination $ax+by=(ax_1+by_1,...,ax_n+by_n)$ belongs to $R^{n}.$ From earlier exercises we know that $\{ ax:a\in R\}$ and $\{ by:b\in R\}$ are straight lines drawn through the vectors $x,y.$ Therefore $\{ ax+by:a,b\in R\}$ is a plane drawn through those vectors. We can say that $R^n$ is a linear space in the following sense: for any $x,y\in R^n$ , the whole plane drawn through $x,y$ is contained in $R^n.$

Subsets of $R^n$ may have this property.

Example 1. On the plane, take any straight line passing through the origin (the slope doesn't matter). The equation of such a line is

(1) $ax+by=0,$

where at least one of the coefficients $a,b$ is different from zero. If $b\neq 0,$ we can solve the equation for $y$ and get a more familiar form $y=kx.$ If $b=0,$ we get a vertical line $x=0.$ If two points $(x_1,y_1),$ $(x_2,y_2)$ satisfy (1), then you can check that their linear combination $c(x_1,y_1)+d(x_2,y_2)$ also satisfies (1). A straight line that does not pass through the origin is described by $ax+by=c$ with $c\neq 0.$

Example 2. In $R^3$ , take any straight line or any 2D plane passing through the origin. A straight line is described by a system of two equations $a_1x+b_1y+c_1z=0,$ $a_2x+b_2y+c_2z=0$ and a plane is described by one equation $ax+by+cz=0.$ You can do the algebra as above to show that linear combinations of elements of these sets belong to these sets. However, I suggest using the geometric interpretation of linear operations to show that these are examples of subspaces.

Figure 1. A hyperplane is not a subspace

A straight line or a plane that does not pass through the origin is not a subspace, see Figure 1. The straight line $L$ does not pass through the origin. If we take vectors $X,Y$ from it, their sum, found by the parallelogram rule, does not belong to $L$ .

Definition 1. The set $L\subset R^n$ is called a linear subspace of $R^{n}$ if for any $x,y\in L$ it contains the whole plane drawn through $x,y.$ By induction, any linear combination $a_1x^{(1)}+...+a_mx^{(m)}$ of any elements of a subspace belongs to it.

The definition does not exclude the extreme cases $L=\{0\}$ and $L=R^n$ . These two subspaces are called trivial.

Sometimes we are interested in how a subspace is generated.

Definition 2. Take any vectors $x^{(1)},...,x^{(k)}\in R^n$ and consider the set $L$ of all linear combinations

(2) $a_1x^{(1)}+...+a_kx^{(k)}$ with $a_1,...,a_k\in R.$

This set is a linear subspace (because a linear combination of expressions of type (2) is again of this type) and it is called a span of $x^{(1)},...,x^{(k)}.$ We also say that $L$ is spanned by $x^{(1)},...,x^{(k)}$ and use the notation $L=span[x^{(1)},...,x^{(k)}].$