Matrix subsapces

Matrix subspaces:

• What is a subspace?
• A subspace of a matrix is a vector space contained within another vector space(basically vector space inception).
• Vector space:
• A vector space is a space that consists of a set of elements(vectors), a field, and the two operations of scalar multiplication and addition.

• Therefore, when a subspace is within a vector space the subspace also contains elements and the two operations of scalar multiplication and addition that are "closed". Thus preventing the resulting outputs from being pushed out of the subspace; otherwise the very definition of subspace is violated.

Rules:

1. Must contain the zero vector.
2. If the vectors u and v are in the subspace, then so should the sum of u and v (closed under addition).
3. If the vector u is in subspace, so then should the product of u and any scalar(c) (closed under scalar multiplication).

⭐ Please note: That a subspace is a span of vectors, but not every span of vectors is able to classify as a subspace due to the rules stated above.

Example:

1.Upper left is not in the defined subspace because the line extends outside of the defined limits(either via addition or multiplication) 2. The second is within the subspace because nothing falls out of the space. 3. Not a subspace because a point was able to be mapped out; indicating that it is not closed under one of the two(or both) operations.

Image courtesy of 'The guide to everything'

❗Practice❗:

Exercise S.C20 Robert Beezer

Working within the vector space $P_3$ of polynomials of degree 3 or less, determine if $p\left(x\right)=x^3+6x+4$ is in the subspace $W$ below.$$W=⟨\left\{x^3+x^2+x,x^3+2x-6,x^2-5\right\}⟩$$

Solution

Exercise S.T31 Chris Black

Let $P$ be the set of all polynomials, of any degree. The set $P$ is a vector space. Let $F$ be the subset of $P$ consisting of all polynomials with only terms of odd degree. Prove or disprove: the set $F$ is a subspace of $P$.

Solution

Exercise S.C17 Chris Black

Working within the vector space $C^4$, determine if $b=\left[\begin{array}{c}2\\ 1\\ 2\\ 1\end{array}\right]$ is in the subspace $W$,$$W=⟨\left\{\left[\begin{array}{c}1\\ 2\\ 0\\ 2\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 3\\ 1\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\\ 2\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 2\\ 0\end{array}\right]\right\}⟩$$

Solution

Beezer, R. (n.d.). Subspaces. Retrieved October 07, 2017, from http://linear.ups.edu/html/section-S.html  Problem set and solutions

Black, C. (n.d.). Subspaces. Retrieved October 07, 2017, from http://linear.ups.edu/html/section-S.html  Problem set and solutions

Useful applications:

• The use of matrices and their subspaces even make in appearance in my beloved field, biology more specifically in genetic and evolutionary studies. 
• Subspaces (subspace clustering) were utilized to model evolutionary trends and hypotheses between genetic variance and speciation away from relative species that were once related or closely related through an common ancestor. 

• "We propose a new approach based upon a hypothesis that under evolutionary constraint, descendant sequences from a common ancestor share a mathematical subspace, as opposed to ambient space. In particular, we hypothesize that member nucleotide or peptide sequences lie within low dimensions within the ambient space of all sequences. Therefore, the subspace dimensions are a reflection of mutation and time elapsed since the speciation or duplication within the cluster."

Wallace, T., Sekmen, A., & Wang, X. (2015). Application of Subspace Clustering in DNA Sequence Analysis. Journal of Computational Biology, 22(10), 940–952. http://doi.org/10.1089/cmb.2015.0084

•  Reveal the structure of solutions of linear inhomogeneous equations 
• Continuum modeling:
• https://doi.org/10.1103/PhysRevE.90.042816
• Topological Space 
• Another nifty math blog that covers this very in depth topic:
• https://mathstrek.blog/2013/01/25/topology-subspaces/
• Ring RR: Subring or R-SubModule (Ring Ideal)
• Requires understanding of ring theory
• Group G$G$: Subgroup or Submonoid or Subsemigroup
• Population modeling
• Requires understanding of semigroup theory
• Monoid M$M$: Submonoid or Subsemigroup

*All of the italicized applications are out of my current depth of understanding but they are at a level that I hope to one day understand and wholly grasp(fingers crossed).*

Useful videos:

• Basis of subspaces

• Linear subspaces

Useful documents:

• Everything you've ever wanted to know about vector spaces and subspaces

• Vector spaces and fields