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• Functions in higher dimensions
  From this point of view, a ''vector'' is a pair, $PQ$, of locations $P$ and $Q$. ...d directions and angles between them. Our interest here is the ''algebraic operations'' on vectors.
  113 KB (19,680 words) - 00:08, 23 February 2019
• The gradient
  ...tion of two variables in the two main directions. The result is given by a vector called its gradient. ...ndence, we attach this vector to the point it came from? The result is a ''vector field''. It is a function from ${\bf R}^2$ to ${\bf R}^2$ and it is placed
  74 KB (13,039 words) - 14:05, 24 November 2018
• Differential calculus of parametric curves
  *The single dependent variable is ''multi-dimensional'', a point or a vector in ${\bf R}^n$. As we know, the former point, $X$, is the end of the latter vector, $OX$. In either case, this is just a combination of two function of the sa
  130 KB (22,842 words) - 13:52, 24 November 2018
• Vector and complex variables
  We transition to from numbers to ''vectors''. But what are the operations? This is a $2$-dimensional vector.
  113 KB (18,750 words) - 02:33, 10 December 2018
• More on vector spaces
  Besides [[Euclidean space]]es, another important class of examples of [[vector space]]s is... Observation: There are algebraic operations on functions $f \colon {\bf R} \rightarrow {\bf R}$:
  14 KB (2,471 words) - 21:48, 5 September 2011
• Vector spaces: introduction
  ...important, algebraically! The point is that we can perform some algebraic operations with these entities if we define them properly. ...'t use this notation here. From context, it should be clear when this is a vector.
  14 KB (2,238 words) - 17:38, 5 September 2011
• Transformations
  We are to solve a ''vector equation''; i.e., to find these unknown coefficients: ...way to stretch these two vectors so that the resulting combination is the vector on the right:
  46 KB (7,625 words) - 13:08, 26 February 2018
• Multilinear algebra
  *$1$-vector $v_1 \in V$; *$2$-vector $v_1 \wedge v_2$ with $v_1,v_2 \in V$;
  49 KB (8,852 words) - 00:30, 29 May 2015
• Matrices as functions
  ==Vector functions== ...se are [[vector space]]s, ${\bf R}^2$. We just combined $u$ and $v$ in one vector $(u,v)$.
  13 KB (2,187 words) - 22:17, 9 September 2011
• Functions of several variables
  \text{number}& &&&\text{point or vector} \text{point or vector}& &&&\text{number}
  97 KB (17,654 words) - 13:59, 24 November 2018
• Forms vs vector fields and functions
  ==Vector fields vs discrete 1-forms== Let's recall that a [[vector field]] is given if there is a vector attached to each point of the plane:
  8 KB (1,153 words) - 19:42, 17 April 2013
• Parametric curves
  [[Image:motion along a vector.jpg|right]] #$f(t) = v \cdot t$, a motion along a [[vector]] $v \in {\bf R}^n$ (constant speed), then $f(0) = 0, f(1) = 1$.
  34 KB (5,665 words) - 15:12, 13 November 2012
• Vector functions
  We have considered two types of "vector functions": ...too consider the general vector functions, i.e., both input and output are vector:
  28 KB (4,769 words) - 19:42, 18 August 2011
• Identities of vector calculus
  ...d $V,W$ are [[vector field]]s. Again, the ideas for the arrows come from [[vector calculus]]: *[[gradient]]: functions $\mapsto$ vector fields,
  6 KB (879 words) - 13:00, 17 April 2013
• Continuous functions
  ...terpret the proposition in terms of $C[a,b]$ (assume $A:=B:=[a,b]$). These operations make $C[a,b]$ into?... ...and only if it is continuous with respect to each of its variables. And a vector-valued function is continuous if and only if every of its coordinate funct
  42 KB (7,138 words) - 19:08, 28 November 2015
• Quotients of vector spaces
  Given a [[vector space]] $L$ and a subspace $M$. How do we "remove" $M$ from $L$? Unfortunately, $L \setminus M$ isn't a vector space!
  6 KB (1,115 words) - 16:03, 27 August 2015
• Linear functions in Euclidean space
  '''A linear function preserves vector operations:''' '''Vector functions.''' What if we have a [[vector function]]
  23 KB (3,893 words) - 04:43, 15 February 2013
• Homology groups of graphs
  ...k requests. If the joints are ordered, these requests can be recorded in a vector format, coordinate-wise. For example, $(1,0,1,0,…)$ means: flip the first Recall that a function that satisfies this identity “preserves the operations” of the group and such functions are called ''homomorphisms''. The extens
  28 KB (4,685 words) - 17:25, 28 November 2015
• Calculus Illustrated
  15 The arithmetic operations on functions 1 Operations on sets
  16 KB (1,933 words) - 19:50, 28 June 2021
• Discrete forms
  ...know that the discrete differential forms, as cochains, are organized into vector spaces, one for each degree. Let's review this first. If $p,q$ are two forms of the same degree $k$, it is easy to define algebraic operations on them.
  35 KB (6,055 words) - 13:23, 24 August 2015
• Dual space
  from the reals to the vector space of all linear functions. In fact, it's an [[isomorphism]]! More generally, suppose $V$ is a [[vector space]]. Let
  9 KB (1,390 words) - 16:14, 16 June 2014
• Cohomology
  from the reals to the vector space of all linear functions on the reals. In fact, it's an isomorphism! '''Example.''' An illustration of a vector in $v\in V={\bf R}^2$ and a covector in $u\in V^*$ is given below:
  45 KB (6,860 words) - 16:46, 30 November 2015
• Functions of several variables: derivatives and integrals
  *$M=<m_1,...,m_n>$ is the vector of slopes, Then we can say that the vector $N=<m_1,...,m_n,1>$ is perpendicular to this “plane” in ${\bf R}^n$. Th
  42 KB (6,904 words) - 15:15, 30 October 2017
• Calculus on cubical complexes
  ...know that the discrete differential forms, as cochains, are organized into vector spaces, one for each degree. Let's review this first. If $p,q$ are two forms of the same degree $k$, it is easy to define algebraic operations on them.
  36 KB (6,218 words) - 16:26, 30 November 2015
• Discrete parametric curves
  [[Image:motion along a vector.jpg| center]] $f(t) = v \cdot t$, a motion along a vector $v \in {\bf R}^n$ with constant speed $v$. Then the locations are $f(0) = 0
  32 KB (5,426 words) - 21:57, 5 August 2016
• Duality: forms as cochains
  from the reals to the vector space of all linear functions on the reals. In fact, it's an isomorphism! '''Example.''' An illustration of a vector in $v\in V={\bf R}^2$ and a covector in $u\in V^*$ is given below:
  29 KB (4,540 words) - 13:42, 14 March 2016
• Linear algebra
  Any subset $U$ of ${\bf R}^3$ that is closed with respect to the operations of ${\bf R}^3$ is called ''linear subspace'' or simply ''subspace''. Examining each coordinate of this vector equation produces three equations for these numbers:
  27 KB (4,667 words) - 01:07, 19 February 2011
• Multivectors
  Given a [[vector space]] (or a [[module]]) $V$ over a [[field]] (or [[ring]]) $R$, we think *$1$-vector $v_1 \in V$;
  9 KB (1,564 words) - 17:19, 25 August 2013
• Products
  ...aic operations on the product set ${\bf R} \times {\bf R}$ in terms of the operations on either copy of ${\bf R}$, so that we have an isomorphism: We provide the definition of this operation for two arbitrary vector spaces $V$ and $W$ over field $R$.
  44 KB (7,951 words) - 02:21, 30 November 2015
• Multilinear forms
  Given a vector space $V$, how does one ''compute'' the (algebraic) lengths, areas, volumes The set of such $k$-forms over $V$ is denoted by $\Lambda ^k(V)$. It is a [[vector space]].
  18 KB (3,325 words) - 13:32, 26 August 2013
• Homotopy
  ...space'' if it is equipped with a topology with respect to which its vector operations are continuous: '''Exercise.''' Prove that these are topological vector spaces: ${\bf R}^n$, $C[a,b]$. Hint: don't forget about the product topolog
  46 KB (7,846 words) - 02:47, 30 November 2015
• Forms
  Suppose $F$ is our force vector and $D$ is the displacement vector. Then this is what we know about the work of $F$. First, as we just saw, th Then, the work $W$ of force $F$ along vector $D$ is defined to be:
  13 KB (2,459 words) - 03:27, 25 June 2015
• Functions of several variables OLD
  ==Continuity under algebraic operations== See also [[Continuity under algebraic operations]].
  34 KB (5,636 words) - 23:52, 7 October 2017
• Matrices: part 2
  ==Properties of matrix operations== The vector are linearly dependent!
  14 KB (2,302 words) - 19:46, 27 January 2013
• Vectors
  The two operations are addition of two vectors and multiplication of a vector by a scalar. Coordinate free vector algebra:
  791 bytes (106 words) - 19:39, 28 August 2010
• New vector spaces from old
  [[Subspaces of vector spaces]]: Given a [[vector space]] X and a [[subset]] Y of X.
  1 KB (219 words) - 17:03, 25 March 2010
• Internal structure of a vector space: part 1
  Let's give this set the structure of a vector space. Now, $E$, $N$, given, use the operations to get everything else:
  10 KB (1,614 words) - 17:13, 22 May 2012
• Advanced Topology: exercises
  #Show that the set of differential forms is a vector space. ...paces of discrete differential forms for the complex: [[Image:Describe the vector spaces of discrete differential forms for the complex below.png|center]]
  9 KB (1,487 words) - 18:18, 9 May 2013
• Algebra of discrete differential forms
  ==Discrete differential forms as a vector space== ...s easy to define algebraic operations with them that make the set into a [[vector space]].
  6 KB (1,000 words) - 18:30, 22 August 2015
• Cubical calculus
  In light of this approach, let's take a look at the integral theorems of vector calculus. There are many of them and, with at least one for each dimension, ...cochains<!--\index{differential form}-->, as cochains, are organized into vector spaces, one for each degree/dimension. Let's review this first.
  25 KB (4,238 words) - 02:30, 6 April 2016
• Wedge product of continuous forms
  ...e "multi-vectors". For example, the $2$-vectors should be linear on either vector and and be anti-symmetric. ...$3$-dimensional case each $2$-vector $u \wedge v$ corresponds to a regular vector $u \times v$ and that explains the connection discussed above. In the gener
  14 KB (2,417 words) - 18:16, 22 August 2015
• Forms in Euclidean spaces
  *the vector of functions of the location variables and *the vector of the direction variables.
  44 KB (7,778 words) - 23:32, 24 April 2015
• Algebra of differential forms
  The set of such $k$-forms over $R$ is denoted by $\Omega ^k(R)$. It is a vector space. ==The vector space of differential forms==
  11 KB (1,947 words) - 18:14, 22 August 2015
• Homotopy and homotopy equivalence
  ...space'' if it is equipped with a topology with respect to which its vector operations are continuous: '''Exercise.''' Prove that these are topological vector spaces: ${\bf R}^n$, $C[a,b]$. Hint: don't forget about the product topolog
  45 KB (7,738 words) - 15:18, 24 October 2015
• Tensor fields
  ''[[Multilinear forms]]'' of order $k$ over a real [[vector space]] $V$ are [[multilinear]] antisymmetric functions that give us oriented lengths, area, volumes, etc. They constitute a vector space
  2 KB (289 words) - 22:11, 15 July 2014
• Linear algebra in elementary calculus
  ...serves as a [[basis]]. The obvious choice of operations makes $\Phi^*$ a [[vector space]]. We want to show $\Phi^*/_{\sim}$ is a vector space too.
  4 KB (604 words) - 15:52, 27 August 2015
• Discrete calculus article
  ...y of shape and [[algebra]] is the study of generalizations of [[arithmetic operations]]. The word ''calculus'' is a [[Latin]] word, meaning originally "small pe ...rem of calculus''' states that differentiation and integration are inverse operations. More precisely, it relates the difference quotients to the Riemann sums. I
  27 KB (4,329 words) - 16:02, 1 September 2019
• Classes of functions
  ...f squaring the input to the next level and combine it with other algebraic operations. A ''quadratic polynomial'' is a function: We have applied the following operations to the ''power functions'', $x,\ x^2,\ x^3,\ …$:
  143 KB (24,052 words) - 13:11, 23 February 2019
• Linear algebra: exercises
  3. Suppose ${\bf C}({\bf R})$ is the vector space of all continuous functions. Let the function $T:{\bf C}({\bf R})\rig *Suppose $V$ is a vector space with operations: $v+w=0$ and $rv=0$ for all $v,w∈V,r∈R$. How many elements does $V$ hav
  5 KB (833 words) - 13:36, 14 March 2018
• Linear subspace
  It is a [[subspace of vector space]]. ...$V$ is a ''linear subspace'' if $L$ is a vector space with respect to the operations of $V$.
  443 bytes (83 words) - 12:50, 21 April 2013
• Solving systems of linear equations
  There is a vector equation behind this system of linear (scalar) equations. Our vector space is ${\bf R}$, $x,y \in {\bf R}$.
  5 KB (802 words) - 01:38, 6 September 2011
• Tangent bundles
  Then the work $W$ of force $F$ along vector $D$ is defined to be ...int. The set of all possible directions at point $A\in V={\bf R}^2$ form a vector space of the same dimension. It is $V_A$, a copy of $V$, attached to each p
  16 KB (2,753 words) - 13:55, 16 March 2016
• Topology of Surfaces by Kinsey
  ...calculus: More good stuff (even though the presentation is a bit sketchy): vector fields, ODEs, differentiable manifolds, etc. ...here, beyond just the definition of continuity: continuity under algebraic operations, the [[Intermediate value Theorem]], [[sequences]], etc.''
  5 KB (725 words) - 12:30, 9 September 2016
• Algebra on graphs
  ...ons: $ra+sb,\ \forall r,s \in {\bf R}$. Then $\{a,b \}$ is a basis of this vector space and the idea becomes: ...ne as follows. The joints are ordered and these requests are recorded in a vector format, coordinate-wise. For instance, $(1,0,1,0, . . .)$ means: flip the f
  36 KB (6,177 words) - 02:47, 21 February 2016
• Calculus I, the discrete version
  What happens to the differential(s) as we perform algebraic operations with the function(s)? The operations of differentiation and integration “cancel” each other as follows.
  64 KB (11,521 words) - 19:48, 22 June 2017
• Means
  ...roup, the group may have no division. If these are points in a subset of a vector space, the set may be non-convex. '''Theorem.''' Suppose $G$ is a set with two binary operations:
  10 KB (1,914 words) - 03:05, 6 November 2018
• Linear operator
  '''A linear function preserves vector operations:''' '''Theorem 1.''' If $M, L$ are [[vector space]]s and $A: M → L$ is a linear operator, then
  849 bytes (143 words) - 14:25, 30 July 2012
• From Calculus to Cohomology by Madsen
  *11. [[degree of map|Degree]], [[linking numbers]] and [[index of vector field]]s *15. [[Fiber bundles]] and [[vector bundles]]
  1 KB (118 words) - 13:51, 5 October 2012
• Differential forms: course
  [[Calculus 3: course|Calculus 3]] (or preferably [[Vector calculus: course|vector calculus]]), [[Introduction to abstract mathematics: course|proofs]], [[lin *[[Algebraic operations with forms and cohomology]]
  3 KB (421 words) - 14:44, 8 February 2013
• Calculus as a part of topology
  *For every vector and a number, what is their scalar product? *For every ''vector'', what is its length?
  13 KB (2,233 words) - 14:41, 20 February 2015
• Products of vector spaces
  ...airs (a,b) of elements in X and Y respectively, as a vector space with the operations '''Theorem.''' (1) The [[basis of vector space|bases]] of L and M form a basis of L×M.
  613 bytes (129 words) - 14:24, 4 April 2010
• Relative topology
  ...that for vector spaces (or groups) the subset needs to be closed under the operations while for topological spaces a subset can ''always'' be seen as a subspace. ...' extension of a linear operator $f:A\to W$ to $F:V\to W$, where $V,W$ are vector spaces and $A$ is a subset of $V$.
  34 KB (6,089 words) - 03:50, 25 November 2015
• Homology of images
  The homology groups of this image are [[vector space]]s generated by these [[cycle]]s, as follows. Consider first the imag ...ond, as we shall see shortly, this approach allows us to perform algebraic operations with cycles.
  2 KB (282 words) - 12:25, 12 August 2015
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  p. 14. "..[[vector calculus:course|vector calculus]] -- which is defined only for up to three spacial [[dimension]]s. ...lot from a vertex to the next. "Therefore, the goal of the most filtering operations is to remove the high frequency noise, while being careful to preserve the
  11 KB (1,663 words) - 16:03, 26 November 2012
• Vector space of infinite sequences
  1. Is this a [[vector space]] in some sense? So the operations are well defined.
  2 KB (319 words) - 02:56, 13 October 2011
• MATLAB
  <p>.. the matrix/vector/tensor core is very elegant and powerful.</p> ...tions, they can’t be analyzed as matrices as they aren’t subject to matrix operations. They are in fact tables.</p>
  4 KB (548 words) - 14:55, 9 October 2010
• Fields related to computer vision
  Let’s start with “dual purpose” image processing tasks. These operations are also image analysis tools: ...r [[Robustness of topology]] (specifically, robustness under morphological operations).</li>
  9 KB (1,379 words) - 01:26, 22 February 2011
• Linear Algebra by Messer
  ...some context for the student. I was glad to see [[quotient space]]s and [[vector field]]s as optional topics. ...is, again, no context. Meanwhile, spending so much time on elementary row operations seems wasteful.
  2 KB (325 words) - 19:45, 7 March 2016
• More metric complexes
  ...sets of $|K|$ form a $\sigma$-algebra, i.e., a collection closed under the operations of complement, countable union, and countable intersection. If, furthermore ...ood as the angle between the direction vector of the former and the normal vector of the latter.
  21 KB (3,445 words) - 13:53, 19 February 2016
• Elementary Linear Algebra -- Spring 2018
  ...Linear Algebra.''' Systems of linear equations, matrices and determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and applications *Outcomes: the student will learn vector algebra and matrix algebra and how to apply them.
  4 KB (573 words) - 13:07, 7 April 2018
• Geometric complexes
  ...sets of $|K|$ form a $\sigma$-algebra, i.e., a collection closed under the operations of complement, countable union, and countable intersection. If, furthermore ...ood as the angle between the direction vector of the former and the normal vector of the latter.
  20 KB (3,354 words) - 17:37, 30 November 2015
• Algebraically closed
  ...]], [[ring]] etc, a subset A of X is ''algebraically closed'' if algebraic operations on the elements of A won't result in elements outside of A. Then A is also a group, vector space, etc.
  349 bytes (54 words) - 01:24, 25 March 2011
• Parametric complexes
  '''Exercise.''' Prove that the two definitions produce isomorphic vector spaces for homology over ${\bf R}$. However, since the persistence can decrease under algebraic operations, the elements of high persistence do not, in general, form a subgroup of th
  45 KB (7,255 words) - 03:59, 29 November 2015
• Subspace of vector space
  ...$V$, any subset $U$ of $V$ that is also a vector space with respect to the operations of $V$ is called ''linear subspace'' or simply ''subspace''.
  200 bytes (33 words) - 01:07, 19 February 2011
• Basics Of Image Processing
  ...ng to images as matrices. The reason is that we will not perform algebraic operations with images. Matrices will appear elsewhere in the context of homology theo ...pixel there assigned 3 integers. One can think of a ''color image'' as a 3-vector attached to each pixel or as 3 tables of integers, etc. For mroe see [[Colo
  5 KB (785 words) - 23:46, 3 May 2008
• Vector space
  There are two operations: addition of two vectors and multiplication of a vector by a scalar. The satisfy the following axioms:
  478 bytes (111 words) - 21:08, 11 September 2009
• Is the intersection of two linear subspaces always a linear subspace?
  ...near subspace iff A is [[algebraically closed|closed]] under the algebraic operations of V. But the intersection of two algebraically closed subsets is closed...
  613 bytes (96 words) - 23:35, 18 November 2010
• Linearity
  Preserving vector operations:
  450 bytes (72 words) - 14:26, 30 July 2012
• Image processing
  ...o images as [[matrices]]. The reason is that we will not perform algebraic operations with images. Matrices will appear elsewhere in the context of [[homology th ...pixel there assigned 3 integers. One can think of a ''color image'' as a 3-vector attached to each pixel or as 3 tables of integers, etc. For more see [[Colo
  5 KB (798 words) - 15:16, 25 November 2010
• Derivative
  [[Image:best affine approximation vector.jpg|right]] In this section we'll see how the derivative is affected by [[vector operations]].
  4 KB (659 words) - 01:47, 30 August 2010
• Applications of differential calculus
  ...sions? Below are the possible types of limits that correspond to algebraic operations: [[image:vector field R1.png| center]]
  59 KB (10,063 words) - 04:59, 21 February 2019