A subspace of a Vector Space is a vector space on its own

Illustration by example

Suppose we have a space $\mathbb R^n$ (e.g. $\mathbb R^2$)

What if we removed one vector? Say, we remove $\bf 0$?

  • The space becomes no longer closed under multiplication by scalar. $\forall \mathbf x: \mathbf 0 \cdot \mathbf x = 0$ which we removed.
  • this is not a vector space - it must be closed under all operations

Another candidate:

  • let's consider the positive quarter of the $x/y$ plane (where $x_1, x_2 > 0$):
  • 76ee20e60bb14133a54723133551d98a.png
  • let's take a vector $\vec x$ from there and multiply it by -1. We no longer stay in this quarter.
  • So this is not a vector space

Any line through the origin:

  • 36680970ea4e49dd8690c9ae3b9f8e84.png
  • is it a vector space?
  • yes. We can take any scalar, and the result will still be on the line
  • if the line is not through the origin, then multiplying by 0 will bring us out of the space - so the origin must be included

So, a subspace of a space should form a space on its own: it should be closed under all possible operations on elements in the subspace

Subspaces of $\mathbb R^2$

  • whole $\mathbb R^2$
  • any line through the origin $\mathbf 0_2$
  • only vector $\mathbf 0_2$

Subspaces of $\mathbb R^3$

  • whole $\mathbb R^3$
  • only vector $\mathbf 0_3$
  • any line through the origin $\mathbf 0_3$
  • any plane through the origin $\mathbf 0_3$

Subspaces from Matrices

For a Matrix there are Four Fundamental Subspaces:

Column Space

Suppose we have a matrix $A \in \mathbb R^{3 \times 2}$

$A = \begin{bmatrix} 1 & 3 \\ 2 & 3 \\ 4 & 1 \\ \end{bmatrix}$

Subspace from columns - $C(A)$ - the Column Space of $A$:

  • we cannot just take the two columns and call it a subspace:
  • it also must include all linear combinations of these columns
  • these linear combinations of two vectors form a plane - a subspace $\mathbb R^2$ in the space $\mathbb R^3$
  • since we include all possible combinations, we're guaranteed to have a subspace
  • cf5432f561ec4f14888e8b376c5f438b.png
  • $v_1$ and $v_2$ are 1st and 2nd columns of $A$ - they form a plane through the origin

Subspace Properties

Take $\mathbb R^3$ and 2 subspaces: $P$ (plane) and $L$ (line)

  • is $P \cup L$ a subspace?
    • $P \cup L$ $\equiv$ all vectors in $P$ or $L$ or both
    • not a subspace: take $v_1 \in P$ and $v_2 \in L$. $v_1 + v_2$ maybe somewhere else - go outside of the union
  • is $P \cap L$ a subspace?
    • $P \cap L \equiv$ vectors in both $P$ and $L$
    • yes (see reasoning below)

$S \cap T$ is a subspace:

  • if $v, w \in S$ then $v + w \in S$ (and all linear combinations)
  • if $v, w \in T$ then $v + w \in T$ (and all linear combinations)
  • then if $v, w \in S \cap T$ then $v + w \in S \cap T$