3Blue1Brown-linear algebra notes

view linear algebra from geometrically 官方双语/合集线性代数的本质 - 系列合集_哔哩哔哩_bilibili

In a coord system, each dimension unit vector like length is 1, any vector in coordinate system can be viewed product of those unit vectors , unit vector called basis , any time vector is depending on current basis

scaling 2 vectors and adding is linear combination of those 2 vectors

span of two of vectors is the set of all possible vectors can be reach with all linear combination of a given pair of vectors

in x-y 2D coordinate system, if and

adding vector not expend span, the add one is "linear depend" with existing ones , is linearly dependent, for all possible , then is linear indedenpent with

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

transformation a fancy word of function, with one input then generate one output , view this as movement change to , for after linear transformation geometrically:

all vector in input space can be transformation in output span, how to represent numerically concisely

vector in input space can be track by basis, if we track how basis transform, then input vector can be match with output vector , is after transformation of , likewise for so

as example , now we see output space basis

so , linear transformation is specify/describe by 4 numbers for arbitrary x y(all input space), we see matrix is transformation specify by column vector as basis

formally linear transformation

multiplying two matrices like geometric meaning of applying one transformation then another

, application is then (right -> left), track where and going, first after applying M2, , since applying then is same ( ), the overall effect is same, which means , so the composite transformation basis are two columns, geometrically matches numerically

we can view naturally why and

2D as example, determinant of transformation: tracking area surround by basis, and all other shapes change the same mount, since grid lines evenly space

when , it squishes all of space onto a line or even a single point checking Det is zero means span of output is squishes small dimensions

if after transformation and remain the virtual order Det is positive, otherwise is negative this natural: at start and stay apart, Det is positive, then towards , Det is closing to 0, after line on Det is 0, at last across , Det is nagetive

3D coordinate system Det is volumn, sign apply right-hand rule, some times computations does not fall within the essence of linear algebra, prove take some effort, but if view geometrically it's natural composite area just same each area multiply, it's not official prove but it's give us inspire what it should like

linear algebra mainly application in graphs, robot, solve linear equations support we have in matrix form , it's identical find original vector that after transformation lands on , if there exist unique solution, if we define reverse transformation of is , transform then back as nothing happen

notice we hind in geometrically as transformation numerically as matrix, then like a coin both side, so we got inverse matrix from inverse transformation

we want solve

we transform 3D space into 2D plane, one dimensional number line, even a point, in that rank specify number of dimensions in output basis

for all possible , the sets of all outputs is column space zero vector always in column space, since linear transformation require "origin remain" rank also view as dimensions in column space full rank imply rank == columns count

for full rank transformation only zero vector lands on itself, not full rank transformation a bunch of vectors land on zero vector

the set of vectors land on origin after transformation is null space/kernal of matrix

as same we have two basis, each use 3 coordinate to descripe

-> LT -> column space is a 2D plane slicing through the origin of of 3D space, it use 3D to describe, but it's not cover all 3D space, it's full rank transformation

full rank: column dimension = input space dimension

, it's geometric meaning is projection one to anther then multiply their length if project vector is opposite from original one the result is negative if both vector are perpenclicular, dot product is zero

why dot production connect with projection

consider some 1x2 matrix transform 2x2 vector into number, which means 2D plane -> 1D number, which has the same calculation of dot project

consider this: in 2D x-y coordinate system project unit vector and in to diagonal line, since this transform is linear, there must exist some 1x2 matrix fulfill this job, which not defined in term of numerical vector or vector dot product, since multiplying 1x2 matrix by a 2D vector is the same thing as turning its side and taking a dot product, this transformation inescapably related to to some 2D vector , so in this way dot product has geometrically projecting

for any 2D vector in original space, , tranform to line of , tracking and , mapping into , ,

for non-unit vector , dot product also multiple some scale factor c

matrix multiply (1x2 as example) is the same thing as tasking a dot product

duality : natural but surprising correspondence between two types of mathematical things

, geometrically

3D: something combines 2 different 3D vector to get a new 3D vector

pre-defined:

why corss product geometrically related #1 #2 #3 properties

when there exists a linear transformation to number line, a vector can be found (dual of that transformation), performing the linear transformation is same thing as taking a dot product

hint of steps:

function , f transform 3D -> number line, it's linear ,

-> there must exist unique matrix describe this transformation,

-> there must exist unique vector(dual) makes

-> compare both side , plugging is way of signaling interpret those coefficients as the coordinates, thus

-> in this way, we connect cross product and dot product, has geometric interoperation

-> cross product has geometric perpendicular to plane determined by and , is area determinded by and

coordinate system function as translate between vectors and sets of number, x, y implicit current basis

standard coordinate system (SC) ; bob 's coordinate system (BC), (use SC basis describe)

function as transform bob' s coordinate into current coordinate system(describe by SC), then apply M 100 times(describe by SC), then reverse into BC(describe by BC)

math equation : , after linear transformation some vector is equal scaled by some factor, the vector is eigenvector, the factor is eigenvalue

eigenvector ' s span it unchanged, it geometrically stretch/squish

application: 3D rotation of axis is eigenvector, it's unchanged durning rotation it's easy think 3D rotation in terms of some axis rotation and an angle rather then thinking 3x3 matrix associated with transformation

eigenvector/eigenvalue in in-depend with coordinate system

, zero vector always hold, the only way it's possible for product of a matrix with a non-zero vector to become zero is if the transformation with that squishes space into lower dimension, thus

3D -> plane, line, origin

if every vector has move, thus no eigenvector exists, rotation , thus no real number solution, since rotation lets every vector left there own span

shear transformation , x-axis vector not move or scale,

eigenvectors may have multiple eigenvalue

eigenbasis: eigenvector lines up with basis

diagonal matrix: all the basis vector is eigenvector, eigenvalue is diagonal value

guaranteed to be diagonal with , base vector just get scaled durning transformation

, use eigen value change basis, form diagonal matrix, then multiple 100 times, reverse to original coordinate system

what's vector

determinant and eigenvectors don't care about the coordinate system, how much a transformation scales area, stay on their own span durning transformation

vector-ish qualities:

axioms for vector

grid lines remain parallel and evenly spaced is geometrically equal to 8-axioms more intuitively, the form of vector doesn't really matter like number 3 , can be 3 persons, 3 cars, 3 things, while add, subtract is same thing

Abstract is price of generality

terms between different context, they mean the same thing

show as example

, slove x, y

Cramer's rule is not most efficient ways of solving equation, but it's more intuitive

det(A) = 0 there may be non solution or many solutions

, unique solution exist

after some transformation T, if , T is orthonormal(正交)

surround with parallelogram area equal to y , this parallelogram after transformation change the same of mount det(A), (parallelogram formed by transformed i-hat and ) = , , for the reason,

it's worthy thing how to work in 3D, volume change the same way

A can be transformed to the reduced row echelon form:

1 -2 0 -1 3

0 0 1 2 -2

0 0 0 0 0

It corresponds to (in x1 - x5, the numbers are subscript):

x1 - 2x2 -x4 + 3x5 = 0

x3 + 2x4 -2x5 = 0

0 = 0

The system has infinitely many solutions:

x1 = 2x2 + x4 -3x5

x2 = arbitrary

x3 = -2x4 + 2x5

x4 = arbitrary

x5 = arbitrary

The solution can be written in vector form:

2

1

0 c2

0

0

+

1

0

-2 c4

1

0

+

-3

0

2

0

1

c5

Therefore the null space has a basis formed by the set {

2

1

0

0

0

,

1

0

-2

1

0

,

-3

0

2

0

1

}

B

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Hilbert空间就是定义了内积的空间，其元素没有任何限制，只要在元素间定义了内积就行

有限维Hilbert空间的特例：通常的几何空间，多项式空间等等

向量空间指的是线性空间，也就是空间中的元素是满足线性关系的，线性空间的特点就是里面有一组基，可以用来表示整个空间。

可以证明，只要是定义了内积，那么元素间就满足了某种线性关系，因此Hilbert空间也可以定义为在线性空间中定义了内积的空间。因此Hilbert空间是一种特殊的线性空间

Space

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For the space beyond Earth's atmosphere, see Outer space For all other uses, see Space (disambiguation)

Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction[1] Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of the boundless four-dimensional continuum known as spacetime In mathematics spaces with different numbers of dimensions and with different underlying structures can be examined The concept of space is considered to be of fundamental importance to an understanding of the universe although disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework

Many of the philosophical questions arose in the 17th century, during the early development of classical mechanics In Isaac Newton's view, space was absolute - in the sense that it existed permanently and independently of whether there were any matter in the space[2] Other natural philosophers, notably Gottfried Leibniz, thought instead that space was a collection of relations between objects, given by their distance and direction from one another In the 18th century, Immanuel Kant described space and time as elements of a systematic framework which humans use to structure their experience

In the 19th and 20th centuries mathematicians began to examine non-Euclidean geometries, in which space can be said to be curved, rather than flat According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space[3] Experimental tests of general relativity have confirmed that non-Euclidean space provides a better model for explaining the existing laws of mechanics and optics

Contents [hide]

1 Philosophy of space

11 Leibniz and Newton

12 Kant

13 Non-Euclidean geometry

14 Gauss and Poincaré

15 Einstein

2 Mathematics

3 Physics

31 Classical mechanics

32 Astronomy

33 Relativity

34 Cosmology

4 Spatial measurement

5 Geography

6 In psychology

7 See also

8 References

Philosophy of space

In the early 11th century Islamic philosopher and physicist, Ibn al-Haytham (also known as Alhacen or Alhazen), discussed space perception and its epistemological implications in his Book of Optics (1021) His experimental proof of the intromission model of vision led to changes in the way the visual perception of space was understood, contrary to the previous emission theory of vision supported by Euclid and Ptolemy In "tying the visual perception of space to prior bodily experience, Alhacen unequivocally rejected the intuitiveness of spatial perception and, therefore, the autonomy of vision Without tangible notions of distance and size for correlation, sight can tell us next to nothing about such things"[4]

Leibniz and Newton

Gottfried LeibnizIn the seventeenth century, the philosophy of space and time emerged as a central issue in epistemology and metaphysics At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, the English physicist-mathematician, set out two opposing theories of what space is Rather than being an entity which independently exists over and above other matter, Leibniz held that space is no more than the collection of spatial relations between objects in the world: "space is that which results from places taken together"[5] Unoccupied regions are those which could have objects in them and thus spatial relations with other places For Leibniz, then, space was an idealised abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete[6] Space could be thought of in a similar way to the relations between family members Although people in the family are related to one another, the relations do not exist independently of the people[7] Leibniz argued that space could not exist independently of objects in the world because that would imply that there would be a difference between two universes exactly alike except for the location of the material world in each universe But since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them According to the principle of sufficient reason, any theory of space which implied that there could be these two possible universes, must therefore be wrong[8]

Isaac NewtonNewton took space to be more than relations between material objects and based his position on observation and experimentation For a relationist there can be no real difference between inertial motion, in which the object travels with constant velocity, and non-inertial motion, in which the velocity changes with time, since all spatial measurements are relative to other objects and their motions But Newton argued that since non-inertial motion generates forces, it must be absolute[9] He used the example of water in a spinning bucket to demonstrate his argument Water in a bucket is hung from a rope and set to spin, starts with a flat surface After a while, as the bucket continues to spin, the surface of the water becomes concave If the bucket's spinning is stopped then the surface of the water remains concave as it continues to spin The concave surface is therefore apparently not the result of relative motion between the bucket and the water[10] Instead, Newton argued, it must be a result of non-inertial motion relative to space itself For several centuries the bucket argument was decisive in showing that space must exist independently of matter

Kant

Immanuel KantIn the eighteenth century the German philosopher Immanuel Kant developed a theory of knowledge in which knowledge about space can be both a priori and synthetic[11] According to Kant, knowledge about space is synthetic, in that statements about space are not simply true by virtue of the meaning of the words in the statement In his work, Kant rejected the view that space must be either a substance or relation Instead he came to the conclusion that space and time are not discovered by humans to be objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences[12]

Non-Euclidean geometry

Spherical geometry is similar to elliptical geometry On the surface of a sphere there are no parallel linesEuclid's Elements contained five postulates which form the basis for Euclidean geometry One of these, the parallel postulate has been the subject of debate among mathematicians for many centuries It states that on any plane on which there is a straight line L1 and a point P not on L1, there is only one straight line L2 on the plane which passes through the point P and is parallel to the straight line L1 Until the 19th century, few doubted the truth of the postulate; instead debate centered over whether it was necessary as an axiom, or whether it was a theory which could be derived from the other axioms[13] Around 1830 though, the Hungarian János Bolyai and the Russian Nikolai Ivanovich Lobachevsky separately published treatises on a type of geometry which does not include the parallel postulate, called hyperbolic geometry In this geometry, there are an infinite number of parallel lines which pass through the point P Consequently the sum of angles in a triangle is less than 180o and the ratio of a circle's circumference to its diameter is greater than pi In the 1850s, Bernhard Riemann developed an equivalent theory of elliptical geometry, in which there are no parallel lines which pass through P In this geometry, triangles have more than 180o and circles have a ratio of circumference to diameter which is less than pi

Type of geometry Number of parallels Sum of angles in a triangle Ratio of circumference to diameter of circle Measure of curvature

Hyperbolic Infinite < 180o > π < 0

Euclidean 1 180o π 0

Elliptical 0 > 180o < π > 0

Gauss and Poincaré

Carl Friedrich GaussAlthough there was a prevailing Kantian consensus at the time, once non-Euclidean geometries had been formalised, some began to wonder whether or not physical space is curved Carl Friedrich Gauss, the German mathematician, was the first to consider an empirical investigation of the geometrical structure of space He thought of making a test of the sum of the angles of an enormous stellar triangle and there are reports he actually carried out a test, on a small scale, by triangulating mountain tops in Germany[14]

Henri PoincaréHenri Poincaré, a French mathematician and physicist of the late 19th century introduced an important insight which attempted to demonstrate the futility of any attempt to discover by experiment which geometry applies to space[15] He considered the predicament which would face scientists if they were confined to the surface of an imaginary large sphere with particular properties, known as a sphere-world In this world, the temperature is taken to vary in such a way that all objects expand and contract in similar proportions in different places on the sphere With a suitable falloff in temperature, if the scientists try to use measuring rods to determine the sum of the angles in a triangle, they can be deceived into thinking that they inhabit a plane, rather than a spherical surface[16] In fact, the scientists cannot in principle determine whether they inhabit a plane or sphere and, Poincaré argued, the same is true for the debate over whether real space is Euclidean or not For him, it was a matter of convention which geometry was used to describe space[17] Since Euclidean geometry is simpler than non-Euclidean geometry, he assumed the former would always be used to describe the 'true' geometry of the world[18]

Einstein

Albert EinsteinIn 1905, Albert Einstein published a paper on a special theory of relativity, in which he proposed that space and time be combined into a single construct known as spacetime In this theory, the speed of light in a vacuum is the same for all observers - which has the result that two events that appear simultaneous to one particular observer will not be simultaneous to another observer if the observers are moving with respect to one another Moreover, an observer will measure a moving clock to tick more slowly than one which is stationary with respect to them; and objects are measured to be shortened in the direction that they are moving with respect to the observer

Over the following ten years Einstein worked on a general theory of relativity, which is a theory of how gravity interacts with spacetime Instead of viewing gravity as a force field acting in spacetime, Einstein suggested that it modifies the geometric structure of spacetime itself[19] According to the general theory, time goes more slowly at places with lower gravitational potentials and rays of light bend in the presence of a gravitational field Scientists have studied the behaviour of binary pulsars, confirming the predictions of Einstein's theories and Non-Euclidean geometry is usually used to describe spacetime

Mathematics

In modern mathematics, spaces are frequently described as different types of manifolds which are spaces that locally approximate to Euclidean space and where the properties are defined largely on local connectedness of points that lie on the manifold

Physics

Classical mechanics

Classical mechanics

Newton's Second Law

History of [hide]Fundamental concepts

Space · Time · Mass · Force

Energy · Momentum

[show]Formulations

Newtonian mechanics

Lagrangian mechanics

Hamiltonian mechanics

[show]Branches

Statics

Dynamics

Kinematics

Applied mechanics

Celestial mechanics

Continuum mechanics

Statistical mechanics

[show]Scientists

Newton · Euler · d'Alembert · Clairaut

Lagrange · Laplace · Hamilton · Poisson

This box: view • talk • edit

Space is one of the few fundamental quantities in physics, meaning that it cannot be defined via other quantities because nothing more fundamental is known at the present On the other hand, it can be related to other fundamental quantities Thus, similar to other fundamental quantities (like time and mass), space can be explored via measurement and experiment

Astronomy

Main article: Astronomy

Astronomy is the science involved with the observation, explanation and measuring of objects in outer space

Relativity

Main article: Theory of relativity

Before Einstein's work on relativistic physics, time and space were viewed as independent dimensions Einstein's discoveries have shown that due to relativity of motion our space and time can be mathematically combined into one object — spacetime It turns out that distances in space or in time separately are not invariant with respect to Lorentz coordinate transformations, but distances in Minkowski space-time along space-time intervals are — which justifies the name

In addition, time and space dimensions should not be viewed as exactly equivalent in Minkowski space-time One can freely move in space but not in time Thus, time and space coordinates are treated differently both in special relativity (where time is sometimes considered an imaginary coordinate) and in general relativity (where different signs are assigned to time and space components of spacetime metric)

Furthermore, from Einstein's general theory of relativity, it has been shown that space-time is geometrically distorted- curved -near to gravitationally significant masses[20]

Experiments are ongoing to attempt to directly measure gravitational waves This is essentially solutions to the equations of general relativity which describe moving ripples of spacetime Indirect evidence for this has been found in the motions of the Hulse-Taylor binary system

Cosmology

Main article: Shape of the universe

Relativity theory lead to the cosmological question of what shape the universe is, and where space came from It appears that space was created in the Big Bang and has been expanding ever since The overall shape of space is not known, but space is known to be expanding very rapidly which is evident due to the Hubble expansion

Spatial measurement

Main article: Measurement

The measurement of physical space has long been important Although earlier societies had developed measuring systems, the International System of Units, (SI), is now the most common system of units used in the measuring of space, and is almost universally used within science

Currently, the standard space interval, called a standard meter or simply meter, is defined as the distance traveled by light in a vacuum during a time interval of exactly 1/299,792,458 of a second This definition coupled with present definition of the second is based on the special theory of relativity, that our space-time is a Minkowski space[citation needed]

Geography

Geography is the branch of science concerned with identifying and describing the Earth, utilizing spatial awareness to try and understand why things exist in specific locations Cartography is the mapping of spaces to allow better navigation, for visualization purposes and to act as a locational device Geostatistics apply statistical concepts to collected spatial data in order to create an estimate for unobserved phenomena

Geographical space is often considered as land, and can have a relation to ownership usage (in which space is seen as property or territory) While some cultures assert the rights of the individual in terms of ownership, other cultures will identify with a communal approach to land ownership, while still other cultures such as Australian Aboriginals, rather than asserting ownership rights to land, invert the relationship and consider that they are in fact owned by the land Spatial planning is a method of regulating the use of space at land-level, with decisions made at regional, national and international levels Space can also impact on human and cultural behavior, being an important factor in architecture, where it will impact on the design of buildings and structures, and on farming

Ownership of space is not restricted to land Ownership of airspace and of waters is decided internationally Other forms of ownership have been recently asserted to other spaces — for example to the radio bands of the electromagnetic spectrum or to cyberspace

Public space is a term used to define areas of land as collectively owned by the community, and managed in their name by delegated bodies; such spaces are open to all While private property is the land culturally owned by an individual or company, for their own use and pleasure

Abstract space is a term used in geography to refer to a hypothetical space characterized by complete homogeneity When modeling activity or behavior, it is a conceptual tool used to limit extraneous variables such as terrain

In psychology

The way in which space is perceived is an area which psychologists first began to study in the middle of the 19th century, and it is now thought by those concerned with such studies to be a distinct branch within psychology Psychologists analyzing the perception of space are concerned with how recognition of an object's physical appearance or its interactions are perceived

Other, more specialized topics studied include amodal perception and object permanence The perception of surroundings is important due to its necessary relevance to survival, especially with regards to hunting and self preservation as well as simply one's idea of personal space

Several space-related phobias have been identified, including agoraphobia (the fear of open spaces), astrophobia (the fear of celestial space) and claustrophobia (the fear of enclosed spaces)

注：实际使用中一般不说左右上下，而是以start、end的位置来定义和描述排列展示方向

无flex的img标签：

设置img(行内元素)为flex布局:

若设置为nowarp(flex-warp的默认值)，由于默认值flax-basis为auto，因此会对元素进行缩放操作，即缩放至适应盒子的尺寸。

而若某个元素无法缩放(例如设置了宽度)则会将溢出。

如果项目的子元素无法缩小，使用nowrap会导致溢出，或者缩小程度还不够小。

basis定义了该元素的布局空白（available space）的基准值。若设定为auto，则会检测该元素(一个felx-item)的宽度并作为basis的值，若未设定宽度，则使用内部元素(如文字、等)的宽度作为basis的值。

确定各个basis后，则会得出空白部分的大小，如上图中总宽度为500px，各个basis均为100px，则空白大小为200px。而对这200px的使用，会通过flex-grow定义。

flex-grow定义了元素的放大(沿主轴方向增长尺寸)情况。flex为0则不进行拉伸，flex为其余正整数，则会通过计算权重将空白部分摊分给各个元素，如第一个元素为2，其余为1，则会将2002/4=100px摊分给a，50px摊分给b，50px摊分给c，试的三个字母各自进行拉伸(只是所处盒子的大小进行拉伸，字母本身不会拉伸变形)。

flex-shrink则定义了元素的缩放情况。那么可以把flex元素flex-shrink属性设置为正整数来缩小它所占空间到flex-basis以下。在计算flex元素收缩的大小时，它的最小尺寸也会被考虑进去，就是说实际上flex-shrink属性可能会和flex-grow属性表现的不一致。

flex的对齐主要是指在cross轴上的对齐方式(主轴或者说main轴可以通过定义宽度、flex-grow flex-shrink flex-basis等进行自定义对齐)。

这个属性的初始值为stretch，这就是为什么flex元素会默认被拉伸到最高元素的高度。实际上，它们被拉伸来填满flex容器 —— 最高的元素定义了容器的高度。

你也可以设置align-items的值为flex-start，使flex元素按flex容器的顶部对齐, flex-end 使它们按flex容器的下部对齐, 或者center使它们居中对齐

justify-content 属性定义了浏览器如何分配顺着父容器主轴的弹性元素之间及其周围的空间。")属性用来使元素在主轴方向上对齐，主轴方向是通过 flex-direction 设置的方向。初始值是flex-start，元素从容器的起始线排列。但是你也可以把值设置为flex-end，从终止线开始排列，或者center，在中间排列

你也可以把值设置为space-between，把元素排列好之后的剩余空间拿出来，平均分配到元素之间，所以元素之间间隔相等。或者使用space-around，使每个元素的左右空间相等。

在实例中尝试下列justify-content属性的值：

112 1 110 7

001-3 变为 001 -3

所以两个基础解系为：(-1 1 0 0)、（-7 0 3 1)

即为 null space的一组基

orthonormal basis

标准正交基；

正交基底；

规范正交基

And an orthonormal basis of the kernel feature space is constructed 

同时构造该核特征空间的一组正交基。

很高兴第一时间为您解答，祝学习进步

如有问题请及时追问，谢谢~~O(∩_∩)O