4.1 Vector Spaces

What is a vector space

• ${\mathbb{R}}^n$ is a vector space

  • its where we can do addition and scalar multiplication like 1.4 Subspace (a subspace is a vector space.)

• In general, a vector space $V$ is a set which has addition and scalar multiplication, for any $\vec{x},\vec{y}\in V$ and $t\in \mathbb{R}$:

  • TLDR: V is closed under linear combinations

• Any vector in a vector space is “equivalent” to a vector in ${\mathbb{R}}^n$

  • we will see later that this vector in ${\mathbb{R}}^n$ acts as the coordinates for the standard basis of the vector space

Ex of a vector space

• the set of mxn matrices, $M_{m,n}$ is a vector space

  • M_2,2

$$\begin{array}{r}2\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]-3\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}-1& 1\\ -3& -1\end{array}\right]\end{array}$$

• the set of polynomials with degree at most n, $P_n$ is a vector space

  • P_2 find, if possible $a,b\in \mathbb{R}$ such that

$$\begin{array}{r}a(1+x+x^2)+b(1+2x^2)=5+3x+7x^2\\ 1:a+b=5\\ x:a=3\\ x^2:a+2b=7\\ \left[\begin{array}{ccc}1& 1& |5\\ 1& 0& |3\\ 1& 2& |7\end{array}\right]⟹RREF⟹\end{array}$$

• Linear Independance

  • in $P_2$ is the set $(x^2+1,x^2+x,x+1)$ a linearly independent set?

    • hint $p_2$ is like ${\mathbb{R}}^3$

$$\begin{array}{r}1+x^2⟹\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right],x+x^2⟹\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right],1+x⟹\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]\\ think,is(\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right])L.I.?\\ \left[\begin{array}{cccc}1& 0& 1& |0\\ 0& 1& 1& |0\\ 1& 1& 0& |0\end{array}\right]⟹REF(\left[\begin{array}{cccc}1& 0& 1& |0\\ 0& 1& 1& |0\\ 0& 0& -2& |0\end{array}\right])\end{array}$$

• Yes it is LI

• Basis

  • find a basis for the following subspace of $M_{2,2}$:

$$\begin{array}{r}S=span(\left[\begin{array}{cc}1& 2\\ 0& -1\end{array}\right],\left[\begin{array}{cc}2& 4\\ 0& -2\end{array}\right],\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right])\\ (we,can,make,it,{\mathbb{R}}^4)\\ \left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 1\\ 0& 0& 1\\ -1& -2& 1\end{array}\right]⟹RREF(\left[\begin{array}{ccc}1& 2& 0\\ 0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right])\end{array}$$

• The only linearly independent and not free variable are the first matrix and the third one!