【线性代数】MIT Linear Algebra Lecture 2: Elimination with matrices
Author｜ Rickyの水果摊
Time ｜ 2022.9.2
Lecture 2: Elimination with matrices
Lecture Info
Instructor: Prof. Gilbert Strang
Course Number: 18.06
Topics: 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
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normal form of elimination
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prerequisites of matrix language
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matrix form of elimination
Active Recall Questions
- How to do row elimination on matrix A A A ?
- What are the differences between A ∗ V c o l A*V_{col} A∗Vcol & V r o w ∗ A V_{row} * A Vrow∗A ? (Hint: Draw figures of their results)
- Given
A
3
∗
3
A_{3*3}
A3∗3, how to construct the elementary/elimination & permutation matrix below ?
- subtract row 1 from row 2 to eliminate A 21 A_{21} A21
- exchange r o w 1 , r o w 2 row_1,row_2 row1,row2 of A A A
Answer
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Omitted
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Figures below are from kenjihiranabe 's excellent repository The-Art-of-Linear-Algebra (Which I highly recommend you to star ⭐️)
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A ∗ V c o l = V n e w c o l A*V_{col}=V_{newcol} A∗Vcol=Vnewcol
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V r o w ∗ A = V n e w r o w V_{row}*A=V_{newrow} Vrow∗A=Vnewrow (This is the prerequisite of matrix language of doing elimination❗️)
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elementary matrices comes from Identity matrix I I I
- E 21 = [ 1 0 0 − 1 1 0 0 0 1 ] E_{21} = \begin{bmatrix} 1&0&0\\ -1&1&0\\ 0&0&1\end{bmatrix} E21=⎣ ⎡1−10010001⎦ ⎤ (Hint: view this process by V r o w ∗ A V_{row}*A Vrow∗A)
- P 21 = [ 0 1 0 1 0 0 0 0 1 ] P_{21} = \begin{bmatrix} 0&1&0\\ 1&0&0\\ 0&0&1\end{bmatrix} P21=⎣ ⎡010100001⎦ ⎤ (Hint: view this process by V r o w ∗ A V_{row}*A Vrow∗A)