【线性代数】MIT Linear Algebra Lecture 2: Elimination with matrices

Author｜ Rickyの水果摊

Time ｜ 2022.9.1

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Lecture 2: Elimination with matrices

Lecture Info

1. Instructor: Prof. Gilbert Strang
2. Course Number: 18.06
3. Topics: Linear Algebra
4. Official Lecture Resource: Resource Index of Linear Algebra

Excellent Notes on GitHub

There are some classic, excellent notes from other authors on GitHub, wihch I highly recommend you to star ⭐️ and read 📖

notes-linear-algebra (A systematic notes written in Chinese)

The-Art-of-Linear-Algebra (Focus on visualization of important concept of Linear Algebra)

Video Link

Lecture 2: Elimination with matrices (bilibili)

Lecture 2: Elimination with matrices (YouTube)

Key Points

1. normal form of elimination

2. prerequisites of matrix language

3. matrix form of elimination

4. elementary / elimination matrix

5. permutation matrix

Active Recall Questions

1. How to do row elimination on matrix $A$ ?
2. What are the differences between $A\ast V_{col}$ & $V_{row}\ast A$ ? (Hint: Draw figures of their results)
3. Describe the process of elimination in matrix language with 1 formula. ❗️(Hint: $A=>U$)
4. What’s the relationship between elementary matrices and permutation matrices ?
5. Given $A_{3\ast 3}$, how to construct the elementary/elimination & permutation matrix below ?
   1. subtract row 1 from row 2 to eliminate $A_{21}$
   2. exchange $ro{w}_1,ro{w}_2$ of $A$

Answer

1. Omitted

2. Figures below are from kenjihiranabe 's excellent repository The-Art-of-Linear-Algebra (Which I highly recommend you to star ⭐️)

   1. $A\ast V_{col}=V_{newcol}$

      

   2. $V_{row}\ast A=V_{newrow}$ (This is the prerequisite of matrix language of doing elimination❗️)

      

3. $E_{m\ast n}Eij\dots E_{31}{E}_{21}A=U$ or $E_{final}\ast A=U$

4. When elementary matrices are used to switch the position of rows of a matrix, we use term “permutation matrix” more than elementary matrix

5. elementary matrices comes from Identity matrix $I$

   1. E 21 = [ 1 0 0 − 1 1 0 0 0 1 ] E_{21} =
      [100−110001]" role="presentation" style="position: relative;">⎡⎣⎢1−10010001⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& 0\\ -1& 1& 0\\ 0& 0& 1\end{array}\right]$$
      E21​=⎣ ⎡​1−10​010​001​⎦ ⎤​ (Hint: view this process by $V_{row}\ast A$)
   2. P 21 = [ 0 1 0 1 0 0 0 0 1 ] P_{21} =
      [010100001]" role="presentation" style="position: relative;">⎡⎣⎢010100001⎤⎦⎥$$\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$
      P21​=⎣ ⎡​010​100​001​⎦ ⎤​ (Hint: view this process by $V_{row}\ast A$)