### Vector spaces: Vector spaces and linear subspaces

### Lines and planes

We now generalize the concepts *line* and *plane* to the context of a vector space.

Line

Let #\vec{p}# and #\vec{v}# be two vectors in a vector space and suppose #\vec{v}\neq \vec{0}#. Then the set of vectors

\[ \vec{p}+\lambda \cdot\vec{v} \quad\text{with }\ \lambda \in \mathbb{R} \]

is called a **line** in the vector space. The vector #\vec{p}# is called a **support vector** of the line and the vector #\vec{v}# a **direction vector**.

The description of the line as #\vec{p}+\lambda\cdot \vec{v}# is called a **parametric representation** of the line (with the **parameter** #\lambda#).

If the vector space is #\mathbb{E}^2# or #\mathbb{E}^3#, then the definitions concur with those of a *parameterized line*. Thus, the notion is indeed a generalization from the Euclidean plane and the Euclidean space to the case of a general vector space.

In the parametric representation only a sum and a scalar product of vectors appear. Therefore it is possible to give this more general definition.

Each vector of the line #l# with parametric representation

\[

\vec{p}+\lambda \cdot\vec{v}

\] can be a support vector of #l#: if #\alpha# is a fixed number, then it is easy to verify that the line #m# given by the parametric representation

\[

\left(\vec{p}+\alpha \cdot\vec{v}\right)+\lambda \cdot\vec{v}

\] is the same line as #l#. After all, by rewriting a vector #\left(\vec{p}+\alpha \cdot\vec{v}\right)+\lambda \cdot\vec{v}

# of #m# as #\vec{p}+\left(\alpha +\lambda\right)\cdot\vec{v}#, we see that the vectors of #m# also belong to #l#, and by rewriting #\vec{p}+\lambda \cdot \vec{v}# as #\left(\vec{p}+\alpha\cdot \vec{v}\right)+(\lambda-\alpha) \cdot\vec{v}#, we observe that each vector of #l# also belongs to #m#.

The direction vector of #l# is uniquely determined up to a scalar multiple.

Plane

Let #\vec{p}#, #\vec{v}#, #\vec{w}# be three vectors in a vector space such that #\vec{v}# is not a scalar multiple of #\vec{w}#, and #\vec{w}# is not a scalar multiple of #\vec{v}#. The collection of vectors

\[\vec{p} +\lambda\cdot \vec{v} +\mu\cdot \vec{w} \phantom{xxx}\text{with}\phantom{xxx} \lambda ,\mu \in \mathbb{R}\]

is called a **plane** in the vector space, with **support vector** #\vec{p}# and **direction vectors** #\vec{v}# and #\vec{w}#.

This description is called a **parametric representation** of the plane (with **parameters** #\lambda# and #\mu#).

If the vector space is #\mathbb{E}^3#, then the definitions correspond to the definitions given previously regarding a *parameterized plane in space*. Thus, we have indeed a generalization from the Euclidean plane and the Euclidean space to the case of a general vector space.

*Later*, we will formulate the conditions regarding the two direction vectors as: #\vec{v}# and #\vec{w}# are linearly independent.

Again, each vector of the plane can be a support vector.

The direction vectors can vary within the linear combinations of #\vec{v}# and #\vec{w}#, in such a way that the requirement that none of the two is a scalar multiple of the other, remains satisfied. This can be seen by use of arguments that are similar to the case of the line. The requirement has the effect that #\vec{v}\neq \vec{0}# and #\vec{w}\neq \vec{0}#.

Thus, lines and planes are sets of the form \[\vec{a}+W =\left\{\vec{a}+\vec{w}\mid \vec{w}\in W\right\}\] for a given vector #\vec{a}# and a linear subspace #W#. In the general case (of arbitrary subspaces #W# instead of just lines and planes, or: an arbitrary number of parameters), this collection is called an **affine subspace**.

Lines and planes have much to do with linear subspaces but are not always linear subspaces themselves:

Lines and planes through the origin

A line or plane is a linear subspace if and only if it contains #\vec{0}#. This is precisely the case if the support vector is a linear combination of the direction vectors.

Lines and planes are determined by two and three vectors, respectively:

Line and plane through a given set of points

Let #\vec{p}#, #\vec{q}#, and #\vec{r}# be three different vectors that are not all three on a line.

- There is a unique line through #\vec{p}# and #\vec{q}#; it has parametric representation #\vec{p}+\lambda\cdot\left(\vec{q}-\vec{p}\right)#.
- There is a unique plane that contains #\vec{p}#, #\vec{q}#, and #\vec{r}#; it has parametric representation #\vec{p} +\lambda\cdot\left( \vec{q}-\vec{p}\right) +\mu\cdot\left( \vec{r}-\vec{p}\right)#.

# \rv{0,0,1}+\lambda \cdot\rv{1,0,7}+\mu \cdot\rv{0,1,-4}#

Take #x=\lambda# , #y=\mu#. Then #z=7\lambda -4 \mu +1#, so #V# consists of the vectors

\[\begin{array}{rcl} \rv{x,y,z}&=&\rv{\lambda,\mu,7\lambda -4\mu +1}\\&=&\rv{0,0,1}+\lambda \cdot\rv{1,0,7}+\mu \cdot\rv{0,1,-4}\end{array} \]

We conclude that #V# is the plane in #\mathbb{R}^3# with position vector #\rv{0,0,1}# and directional vectors #\rv{1,0,7}# and #\rv{0,1,-4}#.

**equation**of the plane #V#. We have converted the equation to a

*parametric representation*of the plane.

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