Chapter 1: Vector Spaces
This chapter will be pretty small, as I don't want to delve into vector spaces here. I will probably treat them separately when I make abstract algebra notes, if need be for those. This chapter will include the definition of a vector space, the definition of the dimension, the definition of the Kernel, the definition of the Image and the dimension theorem, which I will state without proof.
Definition (1.1): A group is a set \(G,*)\) such that \(*:GxG->G\), and:
1. (a*b)*c=a*(b*c), for all a, b, c in G.
2. There exists an element \(e\) in G such that \(a*e=e*a=a\), for all \(a\) in G.
3. For all x in G, there exists a y such that \(x*y=y*x=e\).
Definition (1.2): A group is called commutative/abelian if \(x*y=y*x\).
Definition (1.3): A field is a set \(K,+,*\), such that \(+:KxK->K\), \(*:KxK->K\), such that:
1.\((K,+)\) = abelian group, with 0 as the identity element.
2.\((K-{0},*)\) = group with 1 as the identity element.
3. \(a*(b+c)=a*b+a*c; (a+b)*c=a*c+b*c\).
Definition (1.4): A field is called commutative if * is commutative.
Definition (1.5): Let \(V,+\) be a commutative group, and K a commutative field. \(V/K\) is called a vector space over K if we can define \(*:KxV->V\) such that:
1. \(1*v-v\).
2.\((a*b)v=a*(bv)\), where a and b are elements of K and v is an element of V.
3.\(a(v_1+v_2)=av_1+av_2\).
4.\((a+b)v=av+bv\)
Definition (1.6): Let V' be a subset of V. V' is called a sub-space of V if it is a vector space with the operations induced from V.
Notation: V'≤V.
Theorem (1.7): Let \(V/K\) and V' a subset of V. Then, V'≤V iff for all a and b in K, u and v in V', \(au+bv\) is in V'.
Lemma (1.8): Any intersection of vector sub-spaces is a vector sub-space.
Proof: Let V' be the intersection of the vector spaces \(V_1, V_2, ... V_n\).
Let u,v be in V'. This means that there u and v are in all \(V_i\), so \(au+bv\) belongs to \(V_i\), for all i. Thus, \(au+bv\) belongs to the intersection of \(V_i\), for all i, so it belongs to V', which, by Theorem 1.7, concludes the proof.
Definition (1.9): A set of vectors \(v_1, v_2, ... v_n\) is linearly independent iff, for any \(a_1, a_2,...a_n\) in K, \(a_1v_1+a_2v_2+...+a_nv_n\) is different than the 0 vector.
Definition (1.10): A set of vectors \(v_1, v_2, ... v_n\) is called a spanning set iff every vector \(v\) of the set of vectors can be written as a sum of the form \(a_1v_1+...+a_nv_n=v\).
Definition (1.11): Any linearly independent and spanning set is a base. The cardinal of the base is called the dimension of the vector space.
Definition (1.12): Let \(V_1/K\) and \(V_2/K\) be vector spaces over the same field K. A function \(f\) from \(V_1\) to \(V_2\) is called a morphism if it satisfies the following properties:
1. \(f(x+y)=f(x)+f(y)\), for all x and y in \(V_1\).
2.\(f(ax)=af(x)\), for all a in K and x in \(V_1\).
A bijective morphism is called an isomorphism. Two vector spaces are isomorphic if there exists an isomorphism between them.
Definition (1.13): The Kernel of a morphism is the set \(Ker(f)={[xєV_1|f(x)=0]}\), while the image of a morphism is the set \({[f(x)|xєV_1]}\).
Lemma (1.14): f is injective iff \(Ker(f)={[0]}\).
Proof: Presume \(Ker(f)={0}\). If \(f(x_1)=f(x_2)\), by substracting them, we get that \(f(x_1-x_2)=0\), so \(x_1-x_2\) is in the Kernel, which is composed only of 0. Thus, \(x_1=x_2\). Conversely, presume that, if f weren't injective, \(Ker(f)\) is different from {0}. Of course, 0 is included in the Kernel, but, if \(a\) were also in the Kernel, \(f(a)=f(0)=0\), which contradicts injectivity. Thus, the lemma is proved.
The last result of this chapter is called the dimension theorem. Keep this theorem in mind: we will use it later in the course.
Theorem (1.15): The dimension theorem: Let f be a morphism from \(V_1/K\) to \(V_2/K\), with \(V_1/K\) having finite dimension. Then, the following relationship holds: \(dim(Ker(f))+dim(Im(f))=dim(V_1)\).
Take some time to think about this, to understand it. Try proving that the Kernel is a subspace of \(V_1\), while the Image is a subspace of \(V_2\). As I said, this proof will be important later on, when we will discuss the rank of a matrix.
Chapter 2: Matrices: basic definitions and properties
In this section, I will define what a matrix is, and define a few properties of matrices. Our first result will be to derive the notion of a matrix. For the purpose of this, we shall work on the vector spaces \(K^n/K\), \(K^m/K\), where by \(K^n\) I mean the n-th cartesian product of K. A quick observation tells us that a base of the first vector space is taken to the base of the image of the morphism. Thus, after applying a morphism f, we obtain \(f(x)\) from the elements of the basis of the first vector space.
Definition (2.1): Consider a morphism \(f\) going from \(K^n/K\) to \(K^m/K\). Let \(e_1, e_2... e_n\) be the basis of the first vector space. For any x, \(f(x)=x_1f(e_1)+...+x_nf(e_n)\). Let \(f(e_1)=(a_{11}, a_{21}, ... a_{m1})^T, ... f(e_n)=(a_{1n}, a_{2n}, ... a_{nn})^T\). Thus, \(f(x)\) can be written as \(A*X\), where A is the matrix composed of the \(a_i\)-s, while X is the vector \(x_1, x_2,...x_n)^T\), e.g the coordinates of x in the first vector space, in regard to the basis we chose.. Thus, a matrix is a morphism between vector spaces.
Now that we have defined matrices, we can proceed normally with them. First of all, let us define a minor of a matrix, and then the adjoint matrix. For the purpose of this chapter, we shall work with n*n matrices over the field of complex numbers.
Definition (2.2): A minor \(M_{ij}\) of a matrix is the determinant of the matrix obtained by removing the line i and the column j from the original matrix.
Definition (2.3): The adjoint matrix \(A*\) is the matrix defined as follows: for all i and j, \(a*_{ij}=(-1)^{i+j}M_{ji}\).
Definition (2.4): The determinant of a square matrix A is the volume of the hyper-object constructed with the columns of the matrix; it is also equal to the sum over all permutations p of \(ε(p)a_{1p(1)}a_{2p(2)}...a_{np(n)}\), where \(ε(p)\) is the Levi-Civita symbol of p, which is equal to 1 if the permutation is even, and -1 if the permutation is odd.
Don't worry, no one actually uses that formula. We'll derive an easier formula for the determinant in the next chapter.
Theorem (2.5): For any matrices A in \(M_n(C)\), \(AA*=(A*)A=det(A)I_n\).
Proof: Use theorem 2.9, which will be discussed shortly.
Definition (2.6): A square matrix A is called invertible if there exists a matrix B such that \(AB=BA=I_n\).
Observation: B is unique. (this is left as an exercise for the reader). We call the inverse matrix \(A^{-1}\).
The determinant has the property that \(det(AB)=det(A)det(B)\). Assume that known.
Observation: For a matrix to be invertible, its determinant should be different than 0.
Observation: If a matrix is invertible, then its inverse is \(A^{-1}=(det(A))^{-1}A*\).
Theorem (2.7): An invertible matrix with integer entries that has an integer matrix as inverse has determinant equal to \(+/-1\).
Proof: Since the inverse is also an integer matrix, its determinant is also an integer. Thus, we have that \(A*A^{-1}\) has determinant \(det(A*A^{-1})=det(I_n)=1=det(A)*det(A^{-1})\), and the two matrices have integer determinant by definition 2.4, so they are either both 1 or -1.
Exercise (2.8): Let A and B be square matrices such that \(AB=A+B\). Prove that \(AB=BA\).
Another rather important fact that I want to cover during this chapter is the Laplace determinant formula. It's still pretty ugly, and we will only use it as a means to prove 2.5. I will find an application for it though, don't worry (:
Theorem (2.9) (Laplace determinant formula): Let A be a square matrix with \(n^2\) entries. Let us fix a p between 1 and n, and fix the lines \(i_1, i_2, ... i_p\). Then, the determinant of A is \(det(A)=\sum_{1<=j_1<j_2<...<j_p<=n} (-1)^{i_1+i_2+...+i_p+j_1+j_2+...+j_p}M_{i_1, i_2,...i_p}^{j_1, j_2, ... j_p}N_{i_1, i_2,...i_p}^{j_1, j_2, ... j_p}\), where the first M denotes the minor obtained by removing the lines \(i_1, i_2,...i_p\) and the columns \(j_1, j_2, ... j_p\), while N represents the determinant of the matrix obtained by removing all the other lines and columns except those.
Disgusting, isn't it? Don't worry, this formula is almost never used. This chapter was more of an introduction into matrices; the next chapter will probably be the most important.
Chapter 3: Eigenvalues, eigenvectors and the characteristic polynomial
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