45 Matching Annotations
  1. 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?

    1. Even with this very primitive single neuron, you can achieve 90% accuracy when recognizing a handwritten text image1. To recognize all the digits from 0 to 9, you would need just ten neurons to recognize them with 92% accuracy.

      And here is a Google Colab notebook that demonstrates that

  2. Jul 2021
    1. There is no inactive learning, just as there is no inactive reading.

      This underlies the reason why the acceleration of the industrial revolution has applied to so many areas, but doesn't apply to the acceleration of learning.

      Learning is a linear process.

  3. 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.

  4. Apr 2021
  5. Nov 2020
    1. Linear mixed models are an extension of simple linear models to allow both fixed and random effects, and are particularly used when there is non independence in the data, such as arises from a hierarchical structure
  6. Aug 2020
  7. Jul 2020
  8. Jun 2020
  9. May 2020
  10. Apr 2020
  11. 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.

  12. Dec 2019
    1. "You usually think of an argument as a serial sequence of steps of reason, beginning with known facts, assumptions, etc., and progressing toward a conclusion. Well, we do have to think through these steps serially, and we usually do list the steps serially when we write them out because that is pretty much the way our papers and books have to present them—they are pretty limiting in the symbol structuring they enable us to use. Have you even seen a 'scrambled-text' programmed instruction book? That is an interesting example of a deviation from straight serial presentation of steps.3b6b "Conceptually speaking, however, an argument is not a serial affair. It is sequential, I grant you, because some statements have to follow others, but this doesn't imply that its nature is necessarily serial. We usually string Statement B after Statement A, with Statements C, D, E, F, and so on following in that order—this is a serial structuring of our symbols. Perhaps each statement logically followed from all those which preceded it on the serial list, and if so, then the conceptual structuring would also be serial in nature, and it would be nicely matched for us by the symbol structuring.3b6c "But a more typical case might find A to be an independent statement, B dependent upon A, C and D independent, E depending upon D and B, E dependent upon C, and F dependent upon A, D, and E. See, sequential but not serial? A conceptual network but not a conceptual chain. The old paper and pencil methods of manipulating symbols just weren't very adaptable to making and using symbol structures to match the ways we make and use conceptual structures. With the new symbol-manipulating methods here, we have terrific flexibility for matching the two, and boy, it really pays off in the way you can tie into your work.3b6d This makes you recall dimly the generalizations you had heard previously about process structuring limiting symbol structuring, symbol structuring limiting concept structuring, and concept structuring limiting mental structuring.
  13. 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.
  14. 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.

  15. 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

  16. Oct 2018
  17. 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

  18. Sep 2018
  19. Aug 2018
    1. Viewed from a practice perspective, the distinction be­tween cyclic and linear time blurs because it depends on the observer's point of view and moment of observation. In particular cases, simply shifting the observer's vantage point (e.g., from the corporate suite to the factory floor) or changing the period of observation (e.g., from a week to a year) may make either the cyclic or the linear aspect of ongoing practices more salient.

      Could it be that SBTF volunteers are situating themselves in time as a way to respond to a cyclic/linear tension? or a spatial tension?

    1. Coherent, finalized stories are embedded with alinear structure that aligns with clock time and theGregorian calendar (Gabriel 2000, p. 239). Chronol-ogy and objective time implant these stories withan identifiable past, present and future and a linearcausality that provides a temporal structure (a be-ginning, middle and end with plot and characters).This linearity is tied to the inviolability of sequencedevents that occur within a tensed notion of time where,for example, you cannot have a character seeking re-venge before an original insult has occurred, nor canyou have a punishment for a crime that will be com-mitted later.

      For Gabriel, stories have a linear temporal structure (beginning, middle, end) driven by past, present and future events.

    1. Another strategy in dealing with sui generis time consists in juxtaposing clock time to the various forms of 'social time' and considers the latter as the more 'natural' ones, i.e. closer to subjective perceptions of time, or to the temporality that results from adaptations to seasons or other kinds of natural (biological, environmental) rhythm. This strategy, often couched also in terms of an opposition between 'linear' clock time and 'cyclical' time of natural and social rhythms devalues, or at least ques-tions, the temporality of formal organizations which rely heavily on clock time in fulfilling their coordinative and integrative and controlling functions (Young, 1988; Elchardus, 1988).

      by contrasting social time (as a natural phenomenon) against clock time, allows for a more explicit perspective on linear time (clock) and social rhythms when examining social coordination.

  20. Jul 2018
    1. Since time elapses in a linear fashion and users may switch between tasks during the course of a day, the “elapsed” marbles roll into a track below the storage cylinders.

      This is a Western, industrialized perspective of temporal experience and is not universal.

      Wonder how users respond to the marble representation/metaphor -- does this intuitively make sense to them?

    1. he two patterns—monochronic and polychronic—form a continuum, because polychronicity is the extent to which people prefer to engage in two or more tasks simultaneously, and the complete absence of any simultaneous involvements, engaging tasks one at a time, is the least polychronic position on the continuum.

      Monochronic side of the continuum is linear

      Polychronic side of the continuum is cyclical

      Could Adam's timescape help to further describe this phenomenon? (see Perspectives on time: Zimabrdo + Adam slidedeck)

      linear = spatial, historical, irreversible, tied to a beginning

      cyclical = process, rhythmic, seasonal, bounded, sequential, hopeful (past+future+present)

    1. In contrast to the assumption oftime as linear, with ordered chunks progressing ina straightforward manner, people often negotiate time rhythmically, arranging timein patterns and tempos that do not always co-exist harmoniously.

      Does rhythmic time help to explain some of the tension in crowdsourcing crisis data from non-linear social media streams?

    2. In contrast to the assumption oftime as linear, with ordered chunks progressing ina straightforward manner, people often negotiate time rhythmically, arranging timein patterns and tempos that do not always co-exist harmoniously. In line with earlier CSCW findings [e.g., 4, 9, 45, 46], we term thisrhythmic time, which acknowledges both the rhythmic nature of temporal experience as well a potential disorderliness or ‘dissonance’ when temporal rhythms conflict.Like mosaic time, bringing dissonant rhythms into semi-alignment requires adaptation, work, and patience.

      Rhythmic time definition. Counters the idea of linear time.

      How does this fit (or not) with Reddy's notion of temporal rhythms?

    3. We call this prevailing temporal logic ‘circumscribed time.’ We use this label to highlight the underlying orientation to time as a resource that can, and should, be mastered. A circumscribed temporal logic infers that time should be harnessed into ‘productive’ capacity by approaching it as something that can be chunked, allocated to a single use, experienced linearly, and owned. In turn, the norms of society place the burden on individuals to manage and ‘balance’ time as a steward, optimizing this precious resource by way of control and active manipulation.

      Description of the elements of circumscribed time.

    4. Thedominant temporal logicalso conceptualizestime aslinear. In other words,one chunk of time leads to another in a straight progression. While chunks of time can be manipulated and reordered in the course of a day (or week, or month), each chunk of time has a limited duration and each activity has a beginning and an end. An hour is an hour is an hour, and in the course of a day (or a lifetime) hours stack up like a vector, moving one forward in a straightforward progression.

      Definition of linear time.

      WRT to temporal linguistics, linear time drives moving-ego and moving-time metaphors.

  21. Feb 2017
  22. www.digitalrhetoriccollaborative.org www.digitalrhetoriccollaborative.org
    1. indissolubility

      I'm quite interested in this term, and other contra-structural approaches (such as gutters by, and in, which absence becomes "meaningful"). They as a species suggest something of a "non-linear" logic in which elements resist isolations performed to offer an analysis or critique.

  23. May 2016
  24. Apr 2016
  25. Feb 2016
  26. Oct 2015