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Vector Spaces and Subspaces

What is a vector space?

A vector space is just a set V of vectors on which are defined two operations: addition and multiplication. Note: They don't actually have to be addition and multiplication, they just have to hold for these ten axioms, and addition and multiplication hold.

  1. The sum of vectors u and v, denoted u + v is in V (closure under addition)
  2. u + v = v + u
  3. (u + v) + w = u + (v + w)
  4. There is a zero vector 0 (doesn't actually have to be 0) in V such that u + 0 = u
  5. For each u in V, there is a vector -u in V such that u + (-u) = 0
  6. The scalar multiple of u by c, denoted by cu is in V. (closure under scalar multiplication)
  7. c(u + v) = cu + cv
  8. (c + d)u = cu + du
  9. c(du) = cdu
  10. 1u = u

Basically, showing that it's closed under addition and scalar multiplication makes it a vector space given our normal + and *, because of the properties of real numbers. When the operations aren't orthodox, you must show all 10.

What is a subspace?

A subspace is a subset of a vector space that has three properties

  1. It has the zero vector
  2. It's closed under vector addition (Axiom 1 from before)
  3. It's closed under scalar multiplication (Axiom 6 from before)

So, a subspace is a vector space. Conversely, every vector space is a subspace too! (of itself or larger spaces)

Theorem: Span

MathJax TeX Test Page If $v_1 ... v_p$ are in a vector space V, then Span{$v_1...v_p$} is a subspace of V.

Example

MathJax TeX Test Page Is $\begin{bmatrix}a-b\\b-c\\c-a\\b\end{bmatrix}$ a subspace of $\mathbb{R}^4$? $$\begin{bmatrix}a-b\\b-c\\c-a\\b\end{bmatrix} = a\begin{bmatrix}1\\0\\-1\\0\end{bmatrix} + b\begin{bmatrix}-1\\1\\0\\1\end{bmatrix} + c\begin{bmatrix}0\\-1\\1\\0\end{bmatrix}$$ $$=\text{Span}\left(\begin{bmatrix}1\\0\\-1\\0\end{bmatrix} ,\begin{bmatrix}-1\\1\\0\\1\end{bmatrix} ,\begin{bmatrix}0\\-1\\1\\0\end{bmatrix}\right)$$ By our theorem, it is a subspace.

Null Spaces, Column Spaces, and Linear Transformations

Efficient Way to Calculate Determinants