【线性代数】MIT Linear Algebra Lecture 5: Transposes, permutations, spaces R^n

Author｜ Rickyの水果摊

Time ｜ 2022.9.11

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Lecture 5: Transposes, permutations, spaces $R^n$

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 5: Transposes, permutations, spaces $R^n$ (bilibili)

Lecture 5: Transposes, permutations, spaces $R^n$ (YouTube)

Key Points

1. Transposes and symmetric matrix
2. vector space
3. subspaces
4. column space - $C(A)$

Active Recall Questions

1. If $R$ is a rectangular matrix, then $N=R^T\ast R$ must be a symmetric matrix. Why?
2. What are the different types of vector space in general?
3. What are the basic rules that a vector space have to follow?
4. What are the properties that a vector in ${\mathbb{R}}^n$ have? (Hint: dimension of vectors & value type of each components)
5. List all the possible subspaces of ${\mathbb{R}}^n$
6. What’s the differences between ${\mathbb{R}}^2$ and a subspace (plane) of ${\mathbb{R}}^n$
7. List the vector sets of $line1:y=x,line2:y=-x+1$ in ${\mathbb{R}}^2$ (lines go through the origin or not)
8. What’s the differences between a line and a vector in ${\mathbb{R}}^n$
9. Why the subspaces of ${\mathbb{R}}^n$ like line/plane… have to go through the origin? (Hint: relate to Q7&Q8)
10. Given matrix $A_{m\ast n}$, how to determine $C(A)$ ? (Hint: definition of $C(A)$)
11. What’s the relationship between $C(A)$ and ${\mathbb{R}}^m,{\mathbb{R}}^n$ ?

Answer

1. When talking about symmetry, the first thing need to pop out is $A^T=A$. We apply this formula to $R^{T}R$, $(R^{T}R{)}^T=R^T(R^T{)}^T=R^{T}R$. Hence, $R^{T}R$ is symmetric.
2. “vector” can be couple of things:
   1. vector space
   2. matrix space → $\mathbb{M}$
   3. function space → $\mathbb{F}$
   4. zero space → $\mathbb{Z}$
3. If we take any 2 vectors from vector space $S$, then $x_3=c_1{x}_1+c_2{x}_2$ ($c_1,c_2$ are scalars) is still in that space
4. every vector in ${\mathbb{R}}^n$ has $n$ components, and the value type of each components is real number (That’s where letter R come from)
5. here are all the possibilities:
   1. zero vector with n components
   2. Any line / plane … through the origin
   3. ${\mathbb{R}}^n$
6. The number of components each vector has is different
7. here are the 2 vector set of lines in ${\mathbb{R}}^2$
   1. $line1:y=x\to ((0,0),(1,1),(-1,-1)),\dots$
   2. $line2:y=-x+1\to ((0,1),(1,0),(0.5,0.5))$
8. a vector is formed by an arrow and a point, a line is formed by points of vectors
9. here is the explanation:
   1. Take 2 vectors $v_1,v_2$ in $line2:y=-x+1$, their linear combination $v_3=c_1{v}_1+c_2{v}_2$ is not always in that vector set, and zero vector is also not in that space, so this line is not a subspace
   2. When it turns into $line1:y=x$, the definition of subspace is satisfied, so does other lines that go through the origin. Therefore, we say that subspaces have to go through the origin
10. We do elimination to $A$ to get pivot columns, and span $C(A)$ by those pivot columns
11. $C(A)$ is a subspace of ${\mathbb{R}}^m$