### Vector spaces: Vector spaces and linear subspaces

### The notion of linear subspace

Let #W# be a subset of a vector space #V#. If we take two vectors of #W#, then their sum is a vector of #V#, which does not necessarily lie in #W#. If we want #W# with the addition of #V# to be a vector space again, we must demand that this sum lies in #W#. A similar remark applies to the scalar multiplication of vectors from #W#.

A non-empty subset #W# of a vector space #V# is called a

**linear subspace**of #V# if, for all #\vec{p}#, #\vec{q}\in W# and all scalars #\lambda#, #\mu# the following holds:

\[\lambda\cdot \vec{p} + \mu \cdot \vec{q} \in W\]

- The smallest example is the subset #\{\vec{0}\}# of #V#, which only consists of the zero vector. This linear subspace is called the
**trivial**linear subspace of #V#. - The largest example is the subset #V# of #V#, which consists of all vectors of #V#. All other linear subspaces are called
**proper**.

This requirement that a subset is a linear subspace, guarantees that the structure of a vector space can be found on that subset.

Linear subspaces are vector spaces

If #W# is a linear subspace of #V#, then #W# itself, supplied with the addition and scalar multiplication of #V#, is a vector space.

The homogeneous solution of a system of linear equations with #n# unknowns is a linear subspace of #\mathbb{R}^n#:

The homogeneous solution of a system of linear equations is a linear subspace Consider the following general form of a *homogeneous* system of \(m\) linear equations with \(n\) unknowns \(x_1, \ldots, x_n\): \[\left\{\;\begin{array}{rclllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!0\\ a_{21}x_1 \!\!\!\!&+&\!\!\!\! a_{22}x_2 \!\!&+&\!\! \!\!\cdots \!\!\!\!&+&\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!0\\ \vdots &&\vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n\!\!\!\!&=&\!\!\!\!0\end{array}\right.\] Here, all \(a_{ij}\) with \(1\le i\le m, 1\le j\le n\) are real or complex numbers.

We describe a general vector in #\mathbb{R}^n# using the coordinates #x_1,\ldots,x_n#; so we view #\rv{x_1,\ldots,x_n}# as a vector of #\mathbb{R}^n#. This way, the solutions of the system equations can be viewed as a subset #S# of #\mathbb{R}^n#.

The set of solutions of the homogeneous system is a linear subspace of #\mathbb{R}^n#.

No

The set #W# of solutions of the equation #3\cdot x+4\cdot y-z=-12# in #\mathbb{R}^3# is not a linear subspace of #\mathbb{R}^3# because the zero vector of #\mathbb{R}^3# does not satisfy the equation.

We can see this differently: the vectors #\vec{a} =\rv{-4,0,0}# and #\vec{b} =\rv{0,-3,0}# belong to #W# but #\vec{a}+\vec{b}=\rv{-4,-3,0}# does not, because substituting the coordinate values in the equation leads to #-24=-12#. Therefore, the linear combination #\vec{a}+\vec{b}# of #\vec{a}# and #\vec{b}# does not belong to #W#.

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