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. 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.

Figure 1. A hyperplane is not a subspace

Figure 1. A hyperplane is not a subspace

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 cases 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)}.

In this terminology, the first characterization of the matrix image says that it is spanned by that matrix' columns.

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