Definition and Theorems (Chapter 2)

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Definition. A vector space (or linear space) consists of the following:

  1. a field F of scalars;
  2. a set V of objects, called vectors;
  3. a rule (or operation), called vector addition, which associates with each pair of vectors {\alpha},{\beta} in V a vector \alpha+\beta in V, called the sum of {\alpha} and {\beta}, in such a way that
    ( a ) addition is commutative, \alpha+\beta=\beta+\alpha;
    ( b ) addition is associative, {\alpha}+(\beta +\gamma)=(\alpha+\beta)+\gamma;
    ( c ) there is a unique vector 0 in V, called the zero vector, such that \alpha+0=\alpha for all {\alpha} in V;
    ( d ) for each vector {\alpha} in V there is a unique vector -{\alpha} in V such that {\alpha}+(-{\alpha})=0;
  4. a rule (or operation), called scalar multiplication, which associates with each scalar c in F and vector \alpha in V a vector c{\alpha} in V, called the product of c and \alpha, in such a way that
    ( a ) 1\alpha=\alpha for every {\alpha} in V;
    ( b ) (c_1c_2)\alpha=c_1(c_2\alpha);
    ( c ) c(\alpha+\beta)=c\alpha+c\beta;
    ( d ) (c_1+c_2)\alpha=c_1\alpha+c_2\alpha.

Definition. A vector \beta in V is said to be a linear combination of the vectors {\alpha}_1,\dots,{\alpha}_n in V provided there exist scalars c_1,\dots,c_n in F such that

\displaystyle{\beta=c_1{\alpha}_1+\cdots+c_n{\alpha}_n=\sum_{i=1}^nc_i{\alpha}_i}

Definition. Let V be a vector space over the field F. A subspace of V is a subset W of V which is itself a vector space over F with the operation of vector addition and scalar multiplication on V.

Theorem 1. A non-empty subset W of V is a subspace of V if and only if for each pair of vectors {\alpha},{\beta} in W and each scalar c in F the vector c\alpha+\beta is in W.

Lemma. If A is an m\times n matrix over F and B,C are n\times p matrices over F then
\displaystyle{A(dB+C)=d(AB)+AC,\quad \forall d\in F}

Theorem 2. Let V be a vector space over the field F. Then intersection of any collection of subspaces of V is a subspace of V.

Definition. Let S be a set of vectors in a vector space V. The subspace spanned by S is defined to be the intersection W of all subspaces of V which contains S. When S is a finite set of vectors, S=\{\alpha_1,\alpha_2,\dots,\alpha_n\}, we shall simply call W the subspace spanned by the vectors \alpha_1,\alpha_2,\dots,\alpha_n.

Theorem 3. The subspace spanned by a non-empty subset S of a vector space V is the set of all linear combinations of vectors in S.

Definition. If S_1,S_2,\dots,S_k are subsets of a vector space V, the set of all sums {\alpha}_1+{\alpha_2}+\dots+{\alpha}_k of vectors {\alpha}_i in S_i is called the sum of the subsets S_1,S_2,\dots,S_k and is denoted by S_1+S_2+\dots+S_k or by \sum_{i=1}^kS_i.

Definition. Let V be a vector space over F. A subset S of V is said to be linearly dependent (or simply, dependent) if there exist distinct vectors {\alpha}_1,{\alpha}_2,\dots,\alpha_n in S and scalars c_1,c_2,\dots,c_n in F, not all of which are 0, such that

\displaystyle{c_1{\alpha}_1+\cdots+c_n{\alpha}_n=0}

A set which is not linearly depenent is called linearly independent. If the set S contains only finitely many vectors {\alpha}_1,{\alpha}_2,\dots,\alpha_n, we sometimes say that {\alpha}_1,{\alpha}_2,\dots,\alpha_n are dependent (or independent) instead of saying S is dependent (or independent).

Definition. Let V be a vector space. A basis for V is a linealy independent set of vectors in V which spans the space V. The space V is finite-dimensional if it has a finite basis.

Theorem 4. Let V be a vector space which is spanned by a finite set of vectors {\beta}_1,{\beta}_2,\dots,{\beta}_m. Then any independent set of vectors in V is finite and contains no more than m elements.
Corollary 1. If V is a finite-dimensional vector space, then any two bases of V have the same (finite) number of elements.
Corollary 2. Let V be a finite-dimensional vector space and let n=\dim V. Then
( a ) any subset of V which contains more than n vectors is linearly dependent;
( b ) no subset of V which contains fewer than n vectors can span V.

Theorem 5. If W is a subspace of a finite-dimensional vector space V, every linearly independent subset of W is finite and is part of a (finite) basis for W.
Corollary 1. If W is a proper subspace of a finite-dimensional vector space V, then W is finite-dimensional and \dim W<\dim V.
Corollary 2. In a finite-dimensional vector space V every non-empty linearly independent set of vectors is part of a basis.
Corollary 3. Let A be an n\times n matrix over a field F, and suppose the row vectors of A form a linearly independent set of vectors in F^n. Then A is invertible.

Theorem 6. If W_1 and W_2 are finite-dimensional subspaces of a vector space V, then W_1+W_2 is finite-dimensional and

\displaystyle{\dim W_1+\dim W_2=\dim(W_1\cap W_2)+\dim(W_1+W_2)}

Definition. If V is a finite-dimensional vector space, an ordered basis for V is a finite sequence of vectors which is linearly independent and spans V.

Theorem 7. Let V be an n-dimensional vector space over the field F, and let \mathfrak B and \mathfrak B' be two ordered bases of V. Then there is a unique, necessarily invertible, n\times n matrix P with entries in F such that

\displaystyle{[{\alpha}]_{\mathfrak B}=P[{\alpha}]_{\mathfrak B'}\qquad [{\alpha}]_{\mathfrak B'}=P^{-1}[{\alpha}]_{\mathfrak B}}

for every vector \alpha in V. Then columns of P are given by

\displaystyle{P_j=[{\alpha}_j']_{\mathfrak B},\qquad j=1,\dots,n}

Theorem 8. Suppose P is an n\times n invertilbe matrix over F. Let V be an n-dimensional vector space over F, and let \mathfrak B be an ordered basis of V. Then there is a unique basis \mathfrak B' of V such that

\displaystyle{[{\alpha}]_{\mathfrak B}=P[{\alpha}]_{\mathfrak B'}\qquad [{\alpha}]_{\mathfrak B'}=P^{-1}[{\alpha}]_{\mathfrak B}}

for every vector \alpha in V.

Theorem 9. Row-equivalent matrices have the same row space.

Theorem 10. Let R be a non-zero row-reduced echelon matrix. Then the non-zero row vectors of R form a basis for the row space of R.

Theorem 11. Let m and n be positive integers and let F be a field. Suppose W is a subspace of F^n and \dim W\leq m. Then there is precisely one m\times n row-reduced echelon matrix over F which has W as its row space.
Corollary. Each m\times n matrix A is row-equivalent to one and oonly one row-reduced echelon matrix.
Corollary. Let A and B be m\times n matrices over the field F. Then A and B are row-equivalent if and only if they have the same row space.

Definition and Theorems (Chapter 1)

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Theorem 1. Equivalent systems of linear equations have the same solutions.

Theorem 2. To each elementary row operations \bold{e} there corresponds an elementary row operation \bold{e}_1, of the same type as \bold{e}, such that \bold{e}_1(\bold{e}(A))=\bold{e}(\bold{e}_1(A))=A for each A. In other words, the inverse operation (function) of an elementary row operation exists and is an elementary row operation of the same type.

Definition. If A and B are m\times n matrices over the field F, we say that B is row-equivalent to A if B can be obtained from A by a finite sequence of elementary row operations.

Theorem 3. If A and B are row-equivalent m\times n matrices, the homogeneous systems of linear equations AX=0 and BX=0 have exactly the same solutions.

Definition. An m\times n matrix R is called row-reduced if:
( a ) the first non-zero entry in each non-zero row of R is equal to 1;
( b ) each column of R which contains the leading non-zero entry of some row has all its other entries 0.

Theorem 4. Every m\times n matrix over the field F is row-equivalent to a row-reduced matrix.

Definition. An m\times n matrix R is called a row-reduced echelon matrix if:
( a ) R is row-reduced;
( b ) every row of R which has all its entries 0 occurs below every row which has a non-zero entry;
( c ) if rows 1,\dots,r are the non-zero rows of R, and if the leading non-zero entry of row i occurs in column k_i,i=1,\dots,r, then k_1<k_2<\dots<k_r.

Theorem 5. Every m\times n matrix A is row-equivalent to a row-reduced echelon matrix.

Theorem 6. If A is an m\times n matrix and m<n, then the homogeneous system of linear equations AX=0 has a non-trivial solution.

Theorem 7. If A is an n\times n (square) matrix, then A is row-equivalent to the n\times n identity matrix if and only if the system of equations AX=0 has only the trivial solution.

Definition. Let A be an m\times n matrix over the field F and let B be an n\times p matrix over F. The product AB is the m\times p matrix C whose i,j entry is C_{ij}=\sum_{r=1}^{n}A_{ir}B_{rj}.

Theorem 8. If A,B,C are matrices over the field F such that the products BC and A(BC) are defined, then so are the products AB,(AB)C and

\displaystyle{A(BC)=(AB)C}

Definition. An m\times n matrix is said to be an elementary matrix if it can be obtained from the m\times m identity matrix by emans of a single elementary row operation.

Theorem 9. Let e be an elementary row operation and let E be the m\times m elementary matrix E=e(I). Then for every m\times n matrix A, e(A)=EA.
Corollary. Let A and B be m\times n matrices over the field F. Then B is row-equivalent to A if and only if B=PA, where P is a product of m\times m elementary matrices.

Definition. Let A be an n\times n (square) matrix over the field F. An n\times n matrix B such that BA=I is called a left inverse of A; an n\times n matrix B such that AB=I is called a right inverse of A. IF AB=BA=I, then B is called a two-sided inverse (or inverse) of A and A is said to be invertible.

Theorem 10. Let A and B be n\times n matrices over F.
( i ) If A is invertible, so is A^{-1} and (A^{-1})^{-1}=A.
( ii ) If both A and B are invertible, so is AB and (AB)^{-1}=B^{-1}A^{-1}.
Corollary. A product of invertible matrices is invertible.

Theorem 11. An elementary matrix is invertible.

Theorem 12. If A is an n\times n matrix, the following are equivalent.
( i ) A is invertible.
( ii ) A is row-equivalent to the n\times n identity matrix.
( iii ) A is a product of elementary matrices.
Corollary. If A is an invertible n\times n matrix and if a sequence of elementary row operations reduces A to the identity, then that same sequence of operations when applied to I yields A^{-1}.
Corollary. Let A and B be m\times n matrces. Then B is row-equivalent to A if and only if B=PA where P is an invertible m\times m matrix.

Theorem 13. For an n\times n matrix A, the following are equivalent.
( i ) A is invertible.
( ii ) The homogeneous system AX=0 has only the trivial solution X=0.
( iii ) The system of equations AX=Y has a solution X for each n\times 1 matrix Y.
Corollary. A square matrix with either a left or right inverse is invertible.
Corollary. Let A=A_1A_2{\cdots}A_k where A_1,\dots,A_k are n\times n (square) matrices. Then A is invertible if and only if each A_j is invertible.

Linear Algebra (2ed) Hoffman & Kunze 2.5

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这一节是一个没有习题的阶段性总结，但也相对重要。首先定义了row rank，即row space的维度。然后用比较易懂的方式说明Theorem 9：row-equivalent的矩阵有相同的row space（所以有相同的row rank），Theorem 10解释了row-reduced echelon matrix在描述row space时的重要性，非零行可以直接作为row space的一组基，这主要是由于row-reduced echelon有非常好的特性（线性无关）。Theorem 11是一个很有意思的结论，其说明了小于等于m维的F^n的subspace和m\times n的row-reduced echelon matrix有一个一一对应关系。这一定理的第一个Corollary说明每一个m\times n的matrix都row-equivalent to唯一一个row-reduced echelon matrix，第二个corollary说明row-equivalent 和有相同的row space是等价的。
总结一下，就是以下几个命题都等价：
A和B是row-equivalent
A和B有相同的row space
B=PA,P \text{ is invertible}
AX=0和BX=0的解空间相同（尚未完全证明）