《考研線性代數選講》是2013年清華大學出版社出版的圖書，作者是李麗霞、張蓓、劉秀君。
基本介紹
- 書名：考研線性代數選講
- 作者：李麗霞、張蓓、劉秀君
- ISBN：9787302330752
- 定價：18元
- 出版社：清華大學出版社
- 出版時間：2013-8-12
- 裝幀：平裝
- 版次：1
內容簡介,圖書目錄,
內容簡介
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。
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。
圖書目錄
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
