The picture of Linear algebra
Chapter 1 - Vector spaces
Chapter 2 - Linear transformations and Matrices
Linear algebra provides computational means for many other scientific fields, such as machine learning and physics. Motivated by solving linear equations, mathematicians have developed a theoretical algebraic approach rooted from the definition of vector space and relevant objects.
Definition. \textbf{Definition. } A set VV is called a vector space over a field F\mathbb{F} if it is qualified with two operations
+:V×V→V(x,y)↦x+y,\begin{aligned} + : V\times V &\to V \\ (x,y) &\mapsto x+y,\end{aligned}⋅:F×V→V(c,x)↦cx.\begin{aligned} \cdot : \mathbb{F}\times V &\to V \\ (c,x) &\mapsto cx. \end{aligned}
such that for any x,y,z∈Vx,y,z\in V and a,b∈Fa,b\in\mathbb{F}, the following axioms are satisfied 1. x+y=y+x.2. (x+y)+z=x+(y+z).3. ∃0∈V,x+0=x.4. ∃−x∈V,x+(−x)=0.5. 1x=x.6. (ab)x=a(bx).7. a(x+y)=ax+ay.8. (a+b)x=ax+bx.\begin{aligned} &\textbf{1. } x + y = y + x. \\ &\textbf{2. } (x + y) + z = x + (y + z).\\ &\textbf{3. } \exists 0\in V, x + 0 = x.\\ &\textbf{4. } \exists -x\in V, x + (-x) = 0. \\ &\textbf{5. } 1x = x. \\ &\textbf{6. } (ab)x = a(bx). \\ &\textbf{7. } a(x+y) = ax+ay. \\ &\textbf{8. } (a+b)x = ax+bx. \end{aligned}
Elements of VV are called vectors. Element of F\mathbb{F} are called scalars because they are able to "scale" a vector. Hence the two operators are called vector addition and scalar multiplication. The four first axioms state that a vector space is an abelian group of addtion. The last axioms define rules for scalar multiplication. Similar to sub-structure of other algebraic structures, it is natural to define subspaces of a vector space.
Definition. A subset WW of a vector space VV is subspace of VV if itself is a vector space.
Then we studied specific types of mappings related to vector spaces, namely linear transformations and linear forms.
Definition. A map f:V→Wf:V\to W is called a linear transformation if it satisfies the following conditions 1. f(x+y)=f(x)+f(y).2. f(cx)=cf(x)..\begin{aligned} &\textbf{1. } f(x+y) = f(x)+f(y). \\ &\textbf{2. } f(cx) = cf(x).\\ \end{aligned}. Or equivalently, f(cx+y)=cf(x)+f(y)f(cx+y) = cf(x)+f(y). If W=VW=V, ff is called a linear operator.
A linear transformation between two vector spaces is itself a homomorphism, so a vector space is also called a linear space. We study the matrix representation of a linear transformation and seek an equivalence between a linear transformation and its representation to extend results on linear transformations to matrices. Then we concern the following characteristics of linear transformations
  • Invertibility
  • Diagonalizability
  • Invariance, specifically the eigenvalues and eigenvectors
Definition. A map f:Vk→Ff:V^k\to \mathbb{F} is called a linear form if it satisfies the following conditions 1. f(x1,...,xj+yj,...+xk)=f(x1,...,xj,...,xk)+f(x1,...,yj,...,xk).2. f(x1,...,cxj,...+xk)=cf(x1,...,xj,...,xk).\begin{aligned} &\textbf{1. } f(x_1,...,x_j+y_j,...+x_k) = f(x_1,...,x_j,...,x_k) + f(x_1,...,y_j,...,x_k). \\ &\textbf{2. } f(x_1,...,cx_j,...+x_k) = cf(x_1,...,x_j,...,x_k)\\ \end{aligned}. Or equivalently, (x1,...,cxj+yj,...+xk)=cf(x1,...,xj,...,xk)+f(x1,...,yj,...,xk)(x_1,...,cx_j+y_j,...+x_k) = cf(x_1,...,x_j,...,x_k) + f(x_1,...,y_j,...,x_k).
Now it is sufficient for us to define different vector spaces
  • The quotient space.
  • The vector space L(V,W)\mathcal{L}(V,W) of all linear transformations from VV to WW and equivalently, the vector space of matrices.
  • The dual space L(V,R)\mathcal{L}(V,\mathbb{R}) of all linear operators and analogously, the matrix transpose.
Definition. Applicable properties a linear form ff \\ 1. Conjugating bilinear f(x1,x2)=f(x2,x1)‾ f(x_1, x_2) = \overline{f(x_2,x_1)}. If K=R\mathbb{K} = \mathbb{R}, this becomes symmetry. \\ 2. Positive definite bilinear f(x,x)≥0,  ∀x f(x,x) \ge 0,\,\,\forall x. \\ 3. Alternating multilinear f(x1,...,xi,...,xi,...,xk)=0 f(x_1,...,x_i,...,x_i,...,x_k) = 0. \\ 4. Normalized multilinear f(e1,...,ek)=1 f(e_1,...,e_k) = 1, where {e1,...,ek}\{e_1,...,e_k\} is a basis for VV.
Properties 1 and 2 are used to define the inner product and properties 3 and 4 are used to define the determinant. We finally draw the picture of linear algebra as a summary of its concepts.