考研線性代數選講
作者：李麗霞、張蓓、劉秀君圖書詳細信息：
ISBN：9787302330752定價：18元
印次：1-1
裝幀：平裝
印刷日期：2013-8-12
圖書簡介：
本書基於本科階段線性代數課程教學內容，選講其中涉及考研的部分。全書共分 7講，分別為：行列式的計算，矩陣及其運算，矩陣的初等變換與矩陣的秩，向量組的線性相關性，線性方程組，矩陣的特徵值與特徵向量，矩陣的對角化，以及二次型及其標準形。每講均配有一定數量的練習題，並給出了練習題的解答。本書既可作為考研線性代數選講課程的教材，也可作為考研複習階段和本科學生學習線性代數時的參考書。
前 言
線性代數是考研中數學科目必考的內容，其概念抽象，定理較多，各部分內容頻繁滲透，再加上學時較少，使得許多學生基礎知識掌握不牢。而線性代數的考研題目綜合性較強，有一定的難度，很多同學在複習線性代數內容時感覺困難。因此，我們開設考研線性代數選講課程，幫助學生夯實基礎，拓展知識，了解考研題目的類型與難度，掌握正確的複習方法。本書是編者在多年教學經驗及對碩士研究生入學試題深入研究的基礎上，由近年來考研線性代數選講課程的講義整理而成，選擇了考研中線性代數的主要內容。全書共分 7講，每講包括知識要點、典型例題解析、練習題、練習題參考答案。
知識要點部分：對考研大綱中所要求的知識點做了系統總結，幫助同學理解基本概念、公式和結論，達到融會貫通的目的。對教材中未詳細討論而考研中常用的重要結論做了補充證明，意在掃清學生的知識盲點。
典型例題解析部分：精選了考研題中常見的題型，對各類解題方法做了系統總結，幫助同學梳理解題思路，掌握解題方法和技巧，將知識真正轉化為解題能力的提高。
練習題部分：精選了適量的練習題，幫助同學鞏固所學知識。
練習題參考答案部分：給出了練習題較為詳細的解答，方便學生學習。
無論是知識要點部分，還是典型例題解析部分，都非常重視各部分知識的聯繫與滲透，注重與考研要求的緊密結合。本書可以作為考研複習用書，也可作為本科學生學習線性代數時的參考書。
本書由河北科技大學理學院數學系的李麗霞主編，參加編寫的還有河北科技大學理學院數學系的張蓓、劉秀君。由於編者水平所限，書中難免有不妥之處，懇請批評指正。
編 者
2013年 5月
目 錄
第 1講 行列式的計算 ···························································································1一、知識要點 ··································································································1
1. 行列式的定義 ······················································································1
2. 行列式的性質 ······················································································1
3. 行列式的展開定理 ··············································································2
4. 方陣的行列式的性質 ··········································································3
5. 幾個特殊的行列式 ··············································································3
二、典型例題解析···························································································4
題型一求數字型（尤其是含參數）行列式的值 ································4
題型二 n階行列式的計算 ····································································6
題型三求抽象矩陣的行列式 ·······························································7
練習題 ············································································································10
練習題參考答案 ····························································································12
第 2講 矩陣及其運算 ·························································································15
一、知識要點 ································································································15
1. 矩陣的定義 ························································································15
2. 幾種常用的特殊矩陣 ········································································15
3. 矩陣的運算 ························································································16
4. 分塊矩陣及其運算 ············································································19
5. 可逆矩陣 ···························································································21
6. 正交矩陣 ···························································································25
二、典型例題解析·························································································25
題型一矩陣運算的有關問題 ·····························································25
題型二伴隨矩陣的有關問題 ·····························································28
題型三證明矩陣可逆及求逆矩陣 ······················································28
題型四矩陣方程求解 ·········································································31
練習題 ············································································································33
練習題參考答案 ····························································································35
考研線性代數選講
第 3講 矩陣的初等變換與矩陣的秩 ·································································41
一、知識要點 ································································································41
1. 矩陣的初等變換與初等方陣 ····························································41
2. 矩陣的秩及幾個重要結果 ································································42
3. 矩陣的等價 ························································································45
二、典型例題解析·························································································47
題型一矩陣的初等變換與初等方陣的關係 ······································47
題型二矩陣秩的有關問題 ·································································49
練習題 ············································································································52
練習題參考答案 ····························································································55
第 4講 向量組的線性相關性 ·············································································59
一、知識要點 ································································································59
1. 向量及其運算 ····················································································59
2. 向量組的線性組合與線性相關 ························································60
3. 向量組的等價 ····················································································61
4. 向量組的最 (極)大無關組與向量組的秩 ··········································62
二、典型例題解析·························································································63
題型一向量組線性相關與線性無關的判別、向量組的等價 ···········63
題型二求向量組的秩及最大無關組，並將其餘向量用此最大無關組表示 ·····································································67
練習題 ············································································································69
練習題參考答案 ····························································································71
第 5講 線性方程組 ····························································································76
一、知識要點 ································································································76
1. 線性方程組及其解向量 ····································································76
2. 線性方程組解的判別定理 ································································76
3. 齊次線性方程組的基礎解系 ····························································77
4. 解的性質與通解結構 ········································································78
5. 求非齊次線性方程組 Axb的通解 ················································
.78 二、典型例題解析·························································································79
題型一方程組解的判別 ·····································································79
題型二求解線性方程組 ·····································································80
目錄
題型三利用解的性質及解的結構求通解 ··········································86
題型四兩個方程組的公共解及同解性 ··············································87
題型五判斷一個向量（或向量組）是否可由給定的向量組線性表示 ·································································89
練習題 ············································································································90
練習題參考答案 ····························································································93
第 6講 矩陣的特徵值與特徵向量，矩陣的對角化 ········································ 101
一、知識要點 ······························································································ 101
1. 向量組的正交規範化 ······································································ 101
2. 特徵值與特徵向量的定義 ······························································ 101
3. 特徵值與特徵向量的求法 ······························································ 102
4. 特徵值與矩陣的關係 ······································································ 102
5. 特徵向量的性質 ·············································································· 103
6. 各種運算下的特徵值與特徵向量 ··················································· 103
7. 特殊矩陣的特徵值 ·········································································· 104
8. 矩陣的相似 ······················································································ 104
9. 矩陣的（相似）對角化 ·································································· 105
10. (實)對稱矩陣的(相似)對角化 ························································ 106
二、典型例題解析······················································································· 107
題型一求矩陣的特徵值與特徵向量 ················································ 107
題型二矩陣的相似對角化與實對稱矩陣的相似契約對角化 ········· 108
題型三反求參數問題（包括反求矩陣） ········································ 113
練習題 ·········································································································· 116
練習題參考答案 ·························································································· 118
第 7講 二次型及其標準形 ··············································································· 125
一、知識要點 ······························································································ 125
1. (實)二次型的矩陣表示 ···································································· 125
2. 化二次型為標準形 ·········································································· 126
3. 二次型的規範形 ·············································································· 127
4. 正定二次型與正定矩陣 ·································································· 127
5. 矩陣的契約 ······················································································ 128
6. 矩陣的等價、相似與契約三種關係的對比 ··································· 128
考研線性代數選講
二、典型例題解析······················································································· 128
題型一二次型的標準形與規範形 ···················································· 128
題型二正定性的判別與證明 ··························································· 132
題型三矩陣的等價、相似與契約 ···················································· 134
練習題 ·········································································································· 135
練習題參考答案 ·························································································· 137
