### Vector spaces: Vector spaces and linear subspaces

### Affine subspaces

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.

Affine subspace 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.

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.

Intersection of affine subspaces

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}\]

The solution of a system of linear equations is an affine subspace, which is a linear subspace if the system is homogeneous:

The link between systems of linear equations and affine subspaces

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 \[\vec{c}+W\]

\[\vec{a} = \rv{ 4 , 3 , -3 , 3 } \phantom{xxx}\text{and}\phantom{xxx}\vec{b} = \rv{ -5 , 5 , 4 , 8 } \]

Are the two affine subspaces #\vec{a}+W# and #\vec{b}+W# equal to each other?

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{9,-2,-7,-5}# in this case. We conclude that the equality holds if and only if \[ 1\cdot(9)-1\cdot(-2)+1\cdot(-7)+1\cdot(-5)=0\] The left-hand side has the value #-1#, so the answer is: No.

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