25 Matching Annotations
  1. Dec 2022
  2. Nov 2022
    1. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations.

      A vector space is a mathematical structure that allows us to work with things that have both a magnitude and a direction. This is useful for studying physical quantities like forces and velocity. The concept of vector spaces is important for linear algebra, which is a way of solving systems of linear equations.

    2. n mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

      A vector space is a mathematical structure that allows us to work with things that have both a magnitude and a direction. This is useful for studying physical quantities like forces and velocity. The concept of vector spaces is important for linear algebra, which is a way of solving systems of linear equations.

  3. Feb 2022
  4. Oct 2021
    1. Writing an expression in terms of the trace operator opens up opportunities tomanipulate the expression using many useful identities.

      What does writing an expression using trace operator open up to?

    2. the traceoperator is invariant to the transpose operator:

      What is the trace operator invariant for?

    3. What is the Frobenius Norm of a Matrix?

    4. For example, the trace operator providesan alternative way of writing the Frobenius norm of a matrix:

      The trace operator provides the alternative way of writing which norm of the matrix?

    5. Some operations that aredifficult to specify without resorting to summation notation can be specified usingmatrix products and the trace operator.

      Where the trace operator is useful?

  5. Jun 2021
    1. This also happens to explain intuitively some facts. For instance, the fact that there is no canonical isomorphism between a vector space and its dual can then be seen as a consequence of the fact that rulers need scaling, and there is no canonical way to provide one scaling for space. However, if we were to measure the measure-instruments, how could we proceed? Is there a canonical way to do so? Well, if we want to measure our measures, why not measure them by how they act on what they are supposed to measure? We need no bases for that. This justifies intuitively why there is a natural embedding of the space on its bidual.
    2. The dual is intuitively the space of "rulers" (or measurement-instruments) of our vector space. Its elements measure vectors. This is what makes the dual space and its relatives so important in Differential Geometry, for instance.

      A more intuitive description of why dual spaces are useful or interesting.

  6. Jan 2020
    1. ∣00⟩

      Does this just look like

      [ 1 1 0 0 ]

      as in two |0> smooshed together?

    2. ∥U∣ψ⟩∥2=jkl∑​Ujk∗​ψk∗​Ujl​ψl​

      Lost me here...

    3. T


    4. What does it mean for a matrix UUU to be unitary? It’s easiest to answer this question algebraically, where it simply means that U†U=IU^\dagger U = IU†U=I, that is, the adjoint of UUU, denoted U†U^\daggerU†, times UUU, is equal to the identity matrix. That adjoint is, recall, the complex transpose of UUU:

      Starting to get a little bit more into linear algebra / complex numbers. I'd like to see this happen more gradually as I haven't used any of this since college.

  7. Sep 2019
    1. Time for the red pill. A matrix is a shorthand for our diagrams: A matrix is a single variable representing a spreadsheet of inputs or operations.
  8. Aug 2019
    1. This intentional break from pencil-and-paper notation is meant to emphasize how matrices work. To compute the output vector (i.e. to apply the function), multiply each column of the matrix by the input above it, and then add up the columns (think of squishing them together horizontally).

      read while playing with this: http://matrixmultiplication.xyz/

    2. After months of using and learning about matrices, this is the best gist I've come across.

  9. Jul 2019
    1. One major idea in mathematics is the idea of “closure”. This is the ques-tion: What is the set of all things that can result from my proposed oper-ations? In the case of vectors: What is the set of vectors that can result bystarting with a small set of vectors, and adding them to each other andscaling them? This results in a vector space

      closure in mathematics. sounds similar to domain of a function

  10. Oct 2018
  11. yiddishkop.github.io yiddishkop.github.io
    1. 李宏毅 linear algebra lec7

      Textbook: chapter 1.7


      有没有解 ---> 是不是线性组合 ---> 在不在span中。


    2. 李宏毅 linear algebra lec6: Having solution or Not?

      Textbook: chapter 1.6


      能否找到一个 x 使得 \(Ax=b\) 成立.

      • Linear combination
      • span

      有没有解这个问题非常重要:假设 Linear system 是一个电路,现在老板告诉你这个电路要输出 b 这么大的电流,你能不能找到合适的电压源or电流源,还是根本就找不到?



      A system of linear equations is called consistent if it has one or more solutions。

      只要有解就叫做 consistent.


      A system of linear equations is called inconsistent if its solution set is empty(no solution)

      没有解就叫做 inconsistent.


      Naive 方法:线的交点

      把 system of linear equations 的方程都画成直线,如果他们有交点,那么就是有解,否则无解

      General 方法

      定义引入:Linear Combination

      Given a vector set \(\{u_1,u_2,...,u_k\}\)

      The linear combination of the vectors in the set: \(v=c_1u_1+c_2u_2+...+c_ku_k,\ c_1,c_2,...,c_k\ are\ scalars\ coefficients\ of\ linear\ combination\)

      linear combination is a vector.

      有了 Linear combination 的定义之后,我们再回一下 lec5 篇末讲解的关于 使用 column view of product of matrix and vector 所以我们可以得到的结论是:

      \(Ax\) 其本质就是一个 linear combination, 他是

      • 以 \(x\) 的每一位为 scalar coefficient of linear combination,
      • 以 columns of \(A\) as vectors 作为 vector set engaged in linear combination, 的一个 linear combination


      对于 \(Ax=b\) 是否有解(x是变量)这件事,实际就是在问:b 是否是columns of A的所有可能的线性组合中的一种。





      引入 independent 向量


      引入 反之不反

      非零非平行 ===> 有解;有解 ==X==> 非零非平行。

      引入 span

      vector set 的所有可能的 linear combination (另一个vector set)就是这组 vector set 的 span。

      \(v = c_1u_1+c_2u_2+...+c_ku_k\)

      \(v\) 毫无疑问是一个向量。

      如果我们穷举所有可能的\(c_1,c_2,...,c_k\),他们所得到的向量的集合(vector set \(V\))就是\(x_1,x_2,...,x_k\)的span,同时,\(x_1,x_2,...,x_k\) 叫做 vector set \(V\) 的 generating set.

      引入 generating set

      \(if\ Vector\ set\ V=Span(S),\ then\ V\ is\ Span\ of\ S, also\ S\ is\ a\ generating\ set\ for\ V,\ or\ S\ generates\ V\)

      \(S\) 可以作为一种描述 \(V\) 特性的方法。为什么我们需要这种描述方法呢?因为 \(V\) 作为一个 span,他通常都非常非常的大(一般都是无穷多个),如果我们想要描述这种无穷大(“无穷”都意味着抽象)的向量的集合,最好的方法就是找到一个更具体(“有限”意味着具体)的可联想的“指标” --- generating set --- 这个向量集合是由什么样的向量集合生成的

      相同的向量集(span)可能由不同的向量集(generating set)产生:

      \(S_1=\begin{vmatrix} 1 \\ -1\end{vmatrix}\)



      引入 span of standard vector

      standard vector 其实就是 one-hot encoding vector. 可以见下:

      \(e_1=\begin{vmatrix}1\\0\\0\end{vmatrix}, e_1=\begin{vmatrix}0\\1\\0\end{vmatrix}, e_1=\begin{vmatrix}0\\0\\1\end{vmatrix}\)

      \(span(e_1)=one\ R^1\ in\ R^3\), one axis in 3D-space \(span(e_1,e_2)=one\ R^2\ in\ R^3\), one 2D-space in 3D-space \(span(e_1,e_2,e_3)=R^3\), whole 3D-space.


      • \(Ax=b\) has solution or not?


      • is \(b\) the linear combination of columns of \(A\)?


      • is \(b\) in the \(span\) of the columns of \(A\)?
    3. 李宏毅 linear algebra lec 5



      1. '->' 以下表示线性系统

      2. 符合加法性:x->y ==> x1+x2->y1+y2

      3. 符合乘法(scalar)性:x->y ==> x1k->yk


      再结合一个超级牛逼的观点广义向量 --- 函数也是一种向量。我们就把线性系统是一条直线的观点边界向外扩展了一些:




      1. 加法性:fn->fc ===> fn1 + fn2-> fc1+fc2

      2. 乘法性:fn->fc ===> fn1k->fc1k




      \(vector\ \Rightarrow LinearSystem\ \Rightarrow vector\)

      \(domain\ \Rightarrow LinearSystem\ \Rightarrow co-domain\)


      可以证明的是(in lec3)任何线性系统都可以表示为联立线性等式,也就是说联立等式与线性系统是等价的

      Linear system is equal to System of linear equations.


      1. 矩阵 符合加法/乘法性 所以其为一个线性系统
      2. 联立方程式 符合加法/乘法性 所以其为一个线性系统






      lec5: 两种方式理解 matrix-vector product

      • 可以按看待matrix,正常看法;
      • 可以按看待matrix,把整个matrix看成一个row向量;

      联立方程式 ---> 按列看待matrix的 product of matrix and vector ---> 联立方程式可以写成 Product of matrix and vector. 因为之前说过任何一个线性系统都可以写成联立方程式,那么矩阵就是一个线性系统。

      \(Ax=b\) 中的 \(A\) 就是一个线性系统

    1. 2. 綫性相加(combinations),伸展(span)和單位矢量 l 綫性代數的本質 第二章



      • basis vector \(\hat{i}\)
      • basis vector \(\hat{j}\)
      • adding together two scaled vectors


      \((-5)\hat{i} + (2)\hat{j}\)


      $$ \begin{vmatrix} -5 \\ 2 \end{vmatrix} $$

      what if we choose different basis vectors?


      although \((3.1)\hat{i} + (-2.9)\hat{j} = \(-0.8)\hat{i}+(1.3)\hat{j}\) 但是该向量的实际表示却完全不同:

      $$ \begin{vmatrix} -0.8 \\ 1.3 \end{vmatrix} \neq \begin{vmatrix} 3.1 \\ -2.9 \end{vmatrix} $$

      所以这里需要给出一种关于线性代数的数字表示法\([3.1, -2.9]\)的一个基本条件:每当使用这种表示法时都必须明确单位向量是什么

      span of vectors


      • 如果两个单位向量之间存在夹角那么他们的线性组合形成的向量一定可以覆盖整个平面
      • 如果两个单位向量处在同一个方向(相同or相反)那么他们的线性组合形成的向量只能覆盖这条直线
      • 如果两个单位向量都是 \(\vec{0}\),那么他们的线性组合形成的向量都是\(\vec{0}\)


      The "span" of \(\vec{v}\) and \(\vec{w}\) is the set of all their linear combinations:

      \(a\vec{v} + b\vec{w}\)

      let \(a\) and \(b\) vary over all linear numbers.

      两个向量的 span 与另一个表述是等价的,仅仅通过加法和乘法两种操作可以产生的所有向量

      Vectors VS. Points


      • 那么两个同方向的向量的span就形成一条直线
      • 那么两个不同方向的向量的span就形成一个平面
      • 那么三个不同方向的向量的span就形成一个体

      Redundant and Linearly dependent

      任何时候如果你有多个向量,但是去掉其中一个或几个前者和后者的span没有减少(span is essencially a set --- set of all possible linear combination)


      那么就可以说这个向量与其他向量是 Linear dependent (线性相关), 或者说这个(可以去掉的)向量可以表示为其他向量的线性组合, 因为这个可以去掉的向量处在其他向量的span中

      \(redundant\ \vec{u} \in span(\vec{v}, \vec{w})\)

      或者说,他对扩大span(set of linear combination of vectors)没有作用。

      由此衍生出另一个概念:Linearly independent

      Linearly independent

      \(\vec{u} \neq a\vec{v} + b\vec{w},\ for\ all\ values\ of\ a\ and\ b\)


      basis vector

      有了之前的 span linearly dependent 两个概念,下面才能正式定义第三个概念:何为 basis vector

      The basis of a vector space is a set of linearly independent vectors that span the full space

  12. Sep 2018
  13. Feb 2016