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Definition and Theorems of Chapter 2

2020-02-23 11:04:27 作者：互联网

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

a field $F$ F of scalars;
a set $V$ V of objects, called vectors;
a rule (or operation), called vector addition, which associates with each pair of vectors ${\alpha},{\beta}$ α,β in $V$ V a vector $\alpha+\beta$ α+β in $V$ V, called the sum of ${\alpha}$ α and ${\beta}$ β, in such a way that
( a ) addition is commutative, $\alpha+\beta=\beta+\alpha$ α+β=β+α;
( b ) addition is associative, ${\alpha}+(\beta +\gamma)=(\alpha+\beta)+\gamma$ α+(β+γ)=(α+β)+γ;
( c ) there is a unique vector 0 in $V$ V, called the zero vector, such that $\alpha+0=\alpha$ α+0=α for all ${\alpha}$ α in $V$ V;
( d ) for each vector ${\alpha}$ α in $V$ V there is a unique vector $-{\alpha}$ −α in $V$ V such that ${\alpha}+(-{\alpha})=0$ α+(−α)=0;
a rule (or operation), called scalar multiplication, which associates with each scalar $c$ c in $F$ F and vector $\alpha$ α in $V$ V a vector $c{\alpha}$ cα in $V$ V, called the product of $c$ c and $\alpha$ α, in such a way that
( a ) $1\alpha=\alpha$ 1α=α for every ${\alpha}$ α in $V$ V;
( b ) $(c_1c_2)\alpha=c_1(c_2\alpha)$ (c1c2)α=c1(c2α);
( c ) $c(\alpha+\beta)=c\alpha+c\beta$ c(α+β)=cα+cβ;
( d ) $(c_1+c_2)\alpha=c_1\alpha+c_2\alpha$ (c1+c2)α=c1α+c2α.

Definition. A vector $\beta$ β in $V$ V is said to be a linear combination of the vectors ${\alpha}_1,\dots,{\alpha}_n$ α1,…,αn in $V$ V provided there exist scalars $c_1,\dots,c_n$ c1,…,cn in $F$ F such that
$\beta=c_1{\alpha}_1+\cdots+c_n{\alpha}_n=\sum_{i=1}^nc_i{\alpha}_i$ β=c1α1+⋯+cnαn=i=1∑nciαi

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

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

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

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

Definition. Let $S$ S be a set of vectors in a vector space $V$ V. The subspace spanned by $S$ S is defined to be the intersection $W$ W of all subspaces of $V$ V which contains $S$ S. When $S$ S is a finite set of vectors, $S=\{\alpha_1,\alpha_2,\dots,\alpha_n\}$ S={α1,α2,…,αn}, we shall simply call $W$ W the subspace spanned by the vectors $\alpha_1,\alpha_2,\dots,\alpha_n$ α1,α2,…,αn.

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

Definition. If $S_1,S_2,\dots,S_k$ S1,S2,…,Sk are subsets of a vector space $V$ V, the set of all sums ${\alpha}_1+{\alpha_2}+\dots+{\alpha}_k$ α1+α2+⋯+αk of vectors ${\alpha}_i$ αi in $S_i$ Si is called the sum of the subsets $S_1,S_2,\dots,S_k$ S1,S2,…,Sk and is denoted by $S_1+S_2+\dots+S_k$ S1+S2+⋯+Sk or by $\sum_{i=1}^kS_i$ ∑i=1kSi.

Definition. Let $V$ V be a vector space over $F$ F. A subset $S$ S of $V$ V is said to be linearly dependent (or simply, dependent) if there exist distinct vectors ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$ α1,α2,…,αn in $S$ S and scalars $c_1,c_2,\dots,c_n$ c1,c2,…,cn in $F$ F, not all of which are 0, such that
$c_1{\alpha}_1+\cdots+c_n{\alpha}_n=0$ c1α1+⋯+cnαn=0
A set which is not linearly depenent is called linearly independent. If the set $S$ S contains only finitely many vectors ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$ α1,α2,…,αn, we sometimes say that ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$ α1,α2,…,αn are dependent (or independent) instead of saying $S$ S is dependent (or independent).

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

Theorem 4. Let $V$ V be a vector space which is spanned by a finite set of vectors ${\beta}_1,{\beta}_2,\dots,{\beta}_m$ β1,β2,…,βm. Then any independent set of vectors in $V$ V is finite and contains no more than $m$ m elements.
Corollary 1. If $V$ V is a finite-dimensional vector space, then any two bases of $V$ V have the same (finite) number of elements.
Corollary 2. Let $V$ V be a finite-dimensional vector space and let $n=\dim V$ n=dimV. Then
( a ) any sunset of $V$ V which contains more than $n$ n vectors is linearly dependent;
( b ) no subset of $V$ V which contains fewer than $n$ n vectors can span $V$ V.

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

Theorem 6. If $W_1$ W1 and $W_2$ W2 are finite-dimensional subspaces of a vector space $V$ V, then $W_1+W_2$ W1+W2 is finite-dimensional and
$\dim W_1+\dim W_2=\dim(W_1\cap W_2)+\dim(W_1+W_2)$ dimW1+dimW2=dim(W1∩W2)+dim(W1+W2)

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

Theorem 7. Let $V$ V be an $n$ n-dimensional vector space over the field $F$ F, and let $\mathfrak B$ B and $\mathfrak B'$ B′ be two ordered bases of $V$ V. Then there is a unique, necessarily invertible, $n\times n$ n×n matrix $P$ P with entries in $F$ F such that
$[{\alpha}]_{\mathfrak B}=P[{\alpha}]_{\mathfrak B'}\qquad [{\alpha}]_{\mathfrak B'}=P^{-1}[{\alpha}]_{\mathfrak B}$ [α]B=P[α]B′[α]B′=P−1[α]B
for every vector $\alpha$ α in $V$ V. Then columns of $P$ P are given by
$P_j=[{\alpha}_j']_{\mathfrak B},\qquad j=1,\dots,n$ Pj=[αj′]B,j=1,…,n

Theorem 8. Suppose $P$ P is an $n\times n$ n×n invertilbe matrix over $F$ F. Let $V$ V be an $n$ n-dimensional vector space over $F$ F, and let $\mathfrak B$ B be an ordered basis of $V$ V. Then there is a unique basis $\mathfrak B'$ B′ of $V$ V such that
$[{\alpha}]_{\mathfrak B}=P[{\alpha}]_{\mathfrak B'}\qquad [{\alpha}]_{\mathfrak B'}=P^{-1}[{\alpha}]_{\mathfrak B}$ [α]B=P[α]B′[α]B′=P−1[α]B
for every vector $\alpha$ α in $V$ V.

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

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

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

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