The notions of line, plane, and linear subspace are instances of the following, more general concept. This concept, affine subspace, is the geometric equivalent of a system of linear equations.
Let #V# be a vector space. An affine subspace of #V# is a subset of the form \[\left\{\vec{a}+\vec{w}\mid \vec{w}\in W\right\}\] where #W# is a linear subspace of #V# and #\vec{a}# is a vector of #V#. This affine subspace is often indicated by \[ \vec{a}+W \]
The vector #\vec{a}# is called a support vector of the affine subspace; the subspace #W# is called the direction space of the affine space.
Each linear subspace #W# of #V# is also an affine subspace, as it can be written in the form #\vec{0}+W#.
A line #L# in #V# is defined as the set of vectors of the form #\vec{a}+\lambda\cdot \vec{v}# for fixed vectors #\vec{a}# and #\vec{v}#. The collection #W# of all the scalar multiples of # \vec{v}# is a linear subspace of #V#, so the line #L=\vec{a}+W# is an affine subspace of #V#.
Similarly, a plane #M# in #V# is an affine subspace of #V#: the set #M# consists of all vectors of the form #\vec{a}+\lambda\cdot \vec{v}+\mu\cdot\vec{w}# for fixed vectors #\vec{a}#, #\vec{v}#, and #\vec{w}# (such that #\vec{v}# and #\vec{w}# are not scalar multiples of each other). The collection #U=\lambda\cdot \vec{v}+\mu\cdot\vec{w}# is again a linear subspace of #V#, so the plane #M=\vec{a}+U# is an affine subspace of #V#.
The term support vector of the parametric representation of a line or a plane is consistent with the support vector of the associated affine subspace. So here too the support vector is not unique: each vector in the affine subspace can serve as a support vector.
The direction vectors of a line or plane belong to the linear subspace. Conversely, if #L=\vec{a}+W# is a line, then any vector of #W# which is distinct from #\vec{0}# is a direction vector of #L#; if #M=\vec{a}+U# is a plane, then each pair of vectors of #U# that is not a scalar multiple of each other, is a pair of direction vectors for #M#.
If #\vec{a}# does not belong to #W#, then the affine subspace \(\vec{a}+W\) is not a linear subspace (because #\vec{0}# belongs to #\vec{a}+W# if and only if #\vec{0}=\vec{a}+\vec{w}# for a vector #\vec{w}# of #W#, so if and only if #\vec{w}=-\vec{a}# belongs to #W#; but that is not the case because #\vec{a}# does not belong to #W#).
If two affine subspaces of a vector space have the same direction space, they behave like parallel lines: they coincide or have no point in common. This is a special case of the following theorem.
Let #V# be a vector space, let #U# and #W# be a linear subspaces of #V# and let #\vec{a}# and #\vec{b}# be vectors of #V#. Then we have\[\left(\vec{a}+U\right)\cap \left(\vec{b}+W\right)=\begin{cases}\vec{c}+U\cap W&\text{if there are vectors }\vec{u}\in U\text{ en }\vec{w}\in W\\ &\text{ with }\vec{a}-\vec{b}=\vec{w}-\vec{u}\\&\text{ in which case }\vec{c}=\vec{a}+\vec{u}\\ \emptyset&\text{otherwise}
\end{cases}\]
We will give the proof in the case where #U=W#. The proof of the general case is not much harder and is an exercise.
We use the fact that a vector #\vec{v}# of #V# belongs to #W# if and only if #\vec{v}+W=W#.
This statement can be derived through the following steps, where #\Leftrightarrow# stands for "if and only if" \[\begin{array}{rcl}\vec{v}\in W&\Leftrightarrow &\text{For each } \vec{w}\in W\text{ there is a vector }\vec{u}\text{ in }W\text{ such that }\vec{v}=\vec{w}-\vec{u}\\&\Leftrightarrow &\text{For each } \vec{w}\in W\text{ there is a vector }\vec{u}\text{ in }W\text{ such that }\vec{v}+\vec{u}=\vec{w}\\&\Leftrightarrow &W\subseteq \vec{v}+W\end{array}\] Because #\vec{v}\in W# if and only if #-\vec{v}\in W# it follows from the newly derived statement that also \[\vec{v}\in W\Leftrightarrow W\subseteq -\vec{v}+W\] and therefore \[\vec{v}\in W\Leftrightarrow \vec{v}+W\subseteq W\] We conclude that #\vec{v}\in W# if and only if #W\subseteq\vec{v}+W# and #\vec{v}+W\subseteq W#, which in turn is equivalent to #W=\vec{v}+W#. This proves the fact that we will use.
We now start with the actual proof and assume #\vec{a}-\vec{b}\in W#. Then #\vec{a}-\vec{b}+ W=W#. By adding #\vec{b}# to each element of the sets left and right, we conclude #\vec{a}+W=\vec{b}+W#. This proves the equality of the theorem in case #\vec{a}-\vec{b}\in W#.
For proof of equality in the second case we assume #\vec{a}-\vec{b}\not\in W#. We have to prove that no vector belongs to #\left(\vec{a}+W\right)\cap \left(\vec{b}+W\right)#. Suppose #\vec{v}# lies in this intersection. Then there are vectors #\vec{p}# and #\vec{q}# in #W# such that \[ \vec{v}=\vec{a}+\vec{p}=\vec{b}+\vec{q}\] Rewriting the second equation as \[\vec{a}-\vec{b}=\vec{q}-\vec{p}\] we see that #\vec{a}-\vec{b}# belongs to #W#, because this holds for the linear combination # \vec{q}-\vec{p}# of the two vectors #\vec{p}# and #\vec{q}# in #W#. But this contradicts the assumption #\vec{a}-\vec{b}\not\in W#. We conclude that there is no vector #\vec{v}# in #\left(\vec{a}+W\right)\cap \left(\vec{b}+W\right)#; in other words: this intersection is empty.
If we take #=W#, the theorem gives \[\left(\vec{a}+W\right)\cap \left(\vec{b}+W\right)=\begin{cases}\vec{a}+W&\text{if }\vec{a}-\vec{b}\in W\\ \emptyset&\text{otherwise}\\ \end{cases}\] We see that two affine subspaces with the same direction space either coincide or do not meet at all.
The solution of a system of linear equations is an affine subspace, which is a linear subspace if the system is homogeneous:
Consider the following general form of a system of \(m\) linear equations with \(n\) unknowns \(x_1, \ldots, x_n\): \[\left\{\;\begin{array}{rclllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!b_1\\ a_{21}x_1 \!\!&+&\!\! a_{22}x_2 \!\!&+&\!\! \cdots \!\!&+&\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!b_2\\ \vdots &&\vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n\!\!\!\!&=&\!\!\!\!b_m\end{array}\right.\] Here, all \(a_{ij}\) and \(b_i\) with \(1\le i\le m, 1\le j\le n\) are real numbers.
We point out that this system is called inhomogeneous in general, and homogeneous if the constant terms #b_1,\ldots,b_m# are all equal to #0#. The system of equations which is obtained by replacing the right-hand sides of an inhomogeneous system by zero, is called the associated homogeneous system.
We can interpret the solution of the system of equations in the vector space #\mathbb{R}^n# by regarding #\rv{x_1,\ldots,x_n}# as a general vector of #\mathbb{R}^n#. This way, the solutions of the system of equations can be seen as a subset #S# of #\mathbb{R}^n#.
If #\rv{x_1,\ldots,x_n}=\vec{c}# is a solution of the system of equations and #W# is the linear subspace composed of the solutions of the associated homogeneous system, then the solution of the system is the affine subspace \[\vec{c}+W\]
In theory The notion of linear subspace it is established that the set of solutions of a homogeneous system of linear equations is a linear subspace. This explains why #W# is a linear subspace.
To see why the theorem holds, we treat the case #m=1#. The general case is not essentially more difficult.
The claim comes down to the statement that #\vec{x}=\rv{x_1,\ldots,x_n}# is a solution of \[a_{11}x_1 +a_{12}x_2+ \cdots + a_{1n}x_n=b_1\] if and only if the vector #\vec{w}=\rv{w_1,\ldots,w_n}# determined by #\vec{x}=\vec{c}+\vec{w}# belongs to #W#. This follows from the following stepwise rewriting of the equation in #\vec{x}# to an equation in terms of #\vec{w}#: \[\begin{array}{rcl}a_{11}x_1 +a_{12}x_2+ \cdots + a_{1n}x_n&=&b_1\\ a_{11}\left(c_1+w_1\right)+a_{12}\left(c_2+w_2\right)+ \cdots + a_{1n}\left(c_n+w_n\right)&=&b_1\\ \left( a_{11}c_1+a_{12}c_2+ \cdots + a_{1n}c_n\right)+\left( a_{11}w_1+a_{12}w_2+ \cdots + a_{1n}w_n\right)&=&b_1\\ b_1+\left( a_{11}w_1+a_{12}w_2+ \cdots + a_{1n}w_n\right)&=&b_1\\ a_{11}w_1+a_{12}w_2+ \cdots + a_{1n}w_n&=&0\end{array}\]
Another way to see that the solution of a system of linear equations is an affine subspace, uses the fact that the general solution can be given by a parametric representation \[ \vec{a}+\lambda_1\cdot\vec{v}_1+\cdots+\lambda_k\cdot\vec{v}_k\] with parameters \(\lambda_1,\ldots,\lambda_k\). Such a parametric representation describes an affine subspace: the support vector is the vector #\vec{a}#, and the linear combinations #\lambda_1\cdot\vec{v}_1+\cdots+\lambda_k\cdot\vec{v}_k#, where #\lambda_1,\ldots,\lambda_k# take all possible values, form a linear subspace #W#, as we will see later. Therefore, the parametric representation describes the affine space #\vec{a}+W#.
In the vector space #\mathbb{R}^4# we consider the linear subspace \[W=\left\{\rv{x,y,z,u}\mid x+2\cdot y+3\cdot z- u=0\right\}\] and vectors
\[\vec{a} = \rv{ 2 , 5 , 2 , 22 } \phantom{xxx}\text{and}\phantom{xxx}\vec{b} = \rv{ -4 , 2 , 3 , 12 } \]
Are the two affine subspaces #\vec{a}+W# and #\vec{b}+W# equal to each other?
No
After all, according to the theory, #\vec{a}+W=\vec{b}+W# holds if and only if \[\vec{a}-\vec{b}\in W\] where #\vec{a}-\vec{b}=\rv{6,3,-1,10}# in this case. We conclude that the equality holds if and only if \[ 1\cdot(6)+2\cdot(3)+3\cdot(-1)-1\cdot(10)=0\] The left-hand side has the value #-1#, so the answer is: No.