Linear Algebra: Subspaces

Linear Algebra: Subspaces

Linear Subspaces are subsets of vector spaces that have 3 specific properties. A set of vectors $\Upsilon$ is a subspace if...

1. $\Upsilon$ contains the zero vector $\overrightarrow{0}$
2. Closed under scalar multiplication. $\overrightarrow{x}\in \Upsilon \Rightarrow c\overrightarrow{x}\in \Upsilon \forall <!--FUNCTION\; APPLICATION-->c\in ℝ$.
3. Closed under addition. $\overrightarrow{x}\in \Upsilon \wedge \overrightarrow{y}\in \Upsilon \Rightarrow \overrightarrow{x}+\overrightarrow{y}\in \Upsilon$.

A trivial example of a subspace is just a set $V=\left\{\overrightarrow{0}\right\}$. It contains the zero vector, and any addition and scalar multiplication will just return to the zero vector. What about $S=\left\{\left(x,y\right)\in {ℝ}^2|x>=0\right\}$. This set defines everything in quadrant I and IV, to the right of the y-axis. It contains the zero vector, and any two vectors added together will stay in the subspace. The reason this is not a valid subspace is that scalar multiplication by a negative number can generate vectors outside of the set.

Span and Basis of a Subspace

Span is useful for generating subspaces. A span of linearly dependent vectors will create a line that is a subspace of ${ℝ}^n$. A span of $n$ linearly independent vectors will create a subspace of ${ℝ}^n$ that is also spans ${ℝ}^n$. The basis of a vector space ${ℝ}^n$ is the minimum number of linearly independent vectors whose linear combinations can create all vectors in ${ℝ}^n$. The standard basis of ${ℝ}^2$ is (0, 1) and (1, 0). The number of vectors in the basis is the same as the number of dimensions in the vector space.