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• 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

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