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• Vector and complex variables
  It has two rows and two columns. In other words, this is a $2 \times 2$ matrix. The following combination of $A$ and $B$ is called the ''augmented matrix'' of the system:
  113 KB (18,750 words) - 02:33, 10 December 2018
• Transformations
  It has two rows and two columns. In other words, this is a $2 \times 2$ matrix. This is a $2 \times 1$ matrix.
  46 KB (7,625 words) - 13:08, 26 February 2018
• Functions of several variables: derivatives and integrals
  '''Theorem (Algebra of Derivatives).''' Suppose $f$ and $g$ are differentiable at $X=A$. Then, It has two rows and two columns. In other words, this is a $2 \times 2$ matrix.
  42 KB (6,904 words) - 15:15, 30 October 2017
• Matrices: part 2
  ==Properties of matrix operations== The properties of matrix multiplication are very much like the ones for numbers.
  14 KB (2,302 words) - 19:46, 27 January 2013
• Linear functions in Euclidean space
  There is another connection to linear algebra. The set <center>$dy = f’(a)(dx)$, where $f’(a)$ is a matrix.</center>
  23 KB (3,893 words) - 04:43, 15 February 2013
• Matrices: part 1
  Solve by substitution or by using the "[[augmented matrix]]": *The first matrix is $2 \times 2$: the coefficients of $x,y$.
  8 KB (1,275 words) - 14:18, 9 September 2011
• Metric complexes
  ...obvious that only the way the cells are attached to each other affects the matrix of the boundary operator (and the exterior derivative): In linear algebra, we learn how an inner product adds geometry to a vector space. We choose a
  35 KB (5,871 words) - 22:43, 7 April 2016
• Metric tensor
  ...obvious that only the way the cells are attached to each other affects the matrix of the exterior derivative: However, does this algebra imply concavity? Not without assuming that the intervals have equal lengths
  42 KB (7,131 words) - 17:31, 30 November 2015
• Systems of ODEs
  ...''Hessian matrix'' (discussed in Chapter 18) of $G$. It is the $2\times 2$ matrix of the four partial derivatives of $G$: ...linear function $F$. As such, it is given by a matrix and is evaluated via matrix multiplication:
  63 KB (10,958 words) - 14:27, 24 November 2018
• Linear operators: part 2
  We already know that ''each matrix give rise to a linear operator''. ==How to find a matrix for a linear operator==
  8 KB (1,375 words) - 19:58, 10 September 2011
• PageRank as a solution of a matrix equation
  be the vector of ranks at time $t$. Then, in [[matrix]] notation: where the matrix $M$ is defined as
  5 KB (811 words) - 18:36, 28 November 2012
• Algebra on graphs
  ...is via its ''incidence matrix''<!--\index{incidence matrix}-->, i.e., the matrix with a $1$ in the $(i,j)$-entry if the graph contains edge $ij$ and $0$s el <!--75-->[[image:TopologicalFigure8 and cycle algebra.png| center]]
  36 KB (6,177 words) - 02:47, 21 February 2016
• Determinants of linear operators
  ==How to determine that a matrix is invertible?== Given a matrix or a linear operator $A$, it is either singular or non-singular:
  19 KB (3,177 words) - 18:59, 10 October 2011
• Linear operators: part 3
  ...e left refers to the inverse of the operator while on the right it's about matrix inverse. Note: This is a matrix equation.
  10 KB (1,612 words) - 14:25, 16 October 2013
• Computing homology
  Recall that given a [[cell complex]] $K$, a $k$-[[the algebra of chains|chain]] is a "formal" [[linear combination]] of finitely many ori '''Linear algebra problem.''' Find real numbers (turns out integers) $u, v$, and $w$ such tha
  26 KB (4,370 words) - 21:55, 10 January 2014
• Homology groups of graphs
  ==The algebra of plumbing== ...pursue this analysis via a certain kind of ''algebra''. We introduce this algebra with the following metaphor:
  28 KB (4,685 words) - 17:25, 28 November 2015
• Social choice
  We will look into the first two options as they are subject to the algebra we have developed in this chapter. ==The algebra of vote aggregation==
  47 KB (8,030 words) - 18:48, 30 November 2015
• Vector spaces: introduction
  ==Real numbers and their algebra== Where does this algebra:
  14 KB (2,238 words) - 17:38, 5 September 2011
• Eigenvalues and eigenvectors of linear operators
  Given a basis $\{v_1,v_2\}$, define a linear operator with matrix $A = \left[ If this is not the standard basis, then the matrix is not ''diagonal''.
  12 KB (1,971 words) - 01:09, 12 October 2011
• Linear algebra: exercises
  These are exercises for [[Linear algebra: course]]. [[Linear Algebra by Messer]]
  5 KB (833 words) - 13:36, 14 March 2018
• Inner product
  ...\cdot, \cdot >$ on an $n$-dimensional vector space $V$ can be computed via matrix multiplication where $Q$ is a positive definite, symmetric $n \times n$ matrix.
  4 KB (749 words) - 20:12, 1 May 2013
• Matrices as functions
  Turns out, this is [[matrix product]]! It is called a ''matrix representation'' of this function, $F$. But, if we do have a matrix, we can always understand it as a function, example:
  13 KB (2,187 words) - 22:17, 9 September 2011
• Functions
  ...ften given by ''formulas''. In that case, the above issue is resolved with algebra. ...t: $A$ depends on $w$ only. What is this function? With more middle school algebra, we make this function explicit:
  151 KB (25,679 words) - 17:09, 20 February 2019
• The algebra of oriented cells
  With the algebra we have learned, we can easily conclude the following about the homology of
  36 KB (6,395 words) - 14:09, 1 December 2015
• Homotopy
  as quotients of the maps of chains. However, in comparison, where is the algebra in these homotopy ''groups''? <!--200-->[[image:algebra of loops.png|center]]
  46 KB (7,846 words) - 02:47, 30 November 2015
• Linear operators: part 4
  '''Lesson:''' Linear algebra reveals the ''[[topology]]'' of the graph. ...ifferentiation \hspace{3pt}} \colon {\bf P} \rightarrow {\bf P}$. Find the matrix.
  13 KB (2,067 words) - 01:11, 12 September 2011
• Simplicial homology
  ...clear that another choice of cells' orientations will produce a different algebra of chains... but the same homology groups<!--\index{homology groups}-->! In ==The algebra of oriented chains==
  31 KB (5,170 words) - 13:44, 1 December 2015
• Duality: forms as cochains
  Linear algebra helps one appreciate this seemingly trivial relation. The answer is given b ...cating what $0, -\alpha \in V^*$ are, and then refer to theorems of linear algebra.
  29 KB (4,540 words) - 13:42, 14 March 2016
• Cohomology
  Linear algebra helps one appreciate this seemingly trivial relation. Indeed, the answer is ...cating what $0, -\alpha \in V^*$ are, and then refer to theorems of linear algebra.
  45 KB (6,860 words) - 16:46, 30 November 2015
• Topology of graphs
  ...is via its ''incidence matrix''<!--\index{incidence matrix}-->, i.e., the matrix with a $1$ in the $ij$ position if the graph contains edge $ij$ and $0$s el It is time now to start to recognize the ''need for algebra'' in topology.
  25 KB (4,214 words) - 16:08, 28 November 2015
• Calculus Illustrated
  6 The algebra of exponents 11 The algebra of sums and differences
  16 KB (1,933 words) - 19:50, 28 June 2021
• Test
  #[[Topology vs algebra vs geometry]] #[[Algebra of differential forms]]
  16 KB (2,139 words) - 23:01, 9 February 2015
• Forms in Euclidean spaces
  In order to simplify things, we utilize what we know about the ''algebra'' of directions on ${\bf R}$: the direction from $n$ to $n+1$ is the opposi Both approaches rely on ''the algebra of the Euclidean space''.
  44 KB (7,778 words) - 23:32, 24 April 2015
• College Algebra -- Fall 2016
  '''MTH 130 College Algebra.''' 3 hrs. Polynomial, rational, exponential, and logarithmic functions. Gr *Prerequisites: solid algebra skills, some knowledge of Cartesian coordinates, familiarity with basic fun
  10 KB (1,078 words) - 19:07, 16 December 2016
• Modeling with discrete exterior calculus
  ...geneous case, so for the purposes of speed of the program,the conductivity matrix was replaced with a constant $k=0.6$. ...model is isotropic. In order to do this, we take [[level curve]]s of this matrix at varying times and observe the shape of these curves. We began by taking
  31 KB (5,254 words) - 17:57, 21 July 2012
• Applied Topology and Geometry
  #[[Topology vs algebra vs geometry]] #[[Algebra of differential forms]]
  16 KB (2,088 words) - 16:37, 29 November 2014
• Homotopy and homotopy equivalence
  as quotients of the maps of chains. However, in comparison, where is the algebra in these homotopy ''groups''? [[image:algebra of loops.png|center]]
  45 KB (7,738 words) - 15:18, 24 October 2015
• The algebra of cubical cells
  ...appears twice, it is canceled. The computation is carried out as if we do algebra with ''binary arithmetic''<!--\index{binary arithmetic}-->. That's why we c With the algebra we have learned, we can easily conclude the following about these cycles:
  46 KB (7,844 words) - 12:50, 30 March 2016
• Simplicial complexes and maps
  ==The algebra of oriented chains== Up to this point, the development of the algebra of chains follows the same path as in the case of oriented ''cubical'' comp
  27 KB (4,625 words) - 12:52, 30 March 2016
• College Algebra by Ratti and McWaters
  ''College Algebra'', 2/E by J. S. Ratti and Marcus S. McWaters Used it for [[College Algebra -- Fall 2011]], see also [[College algebra: course]]. It's better than many others I've seen.
  2 KB (269 words) - 18:53, 16 November 2011
• Homology in dimension 2
  Recall that given a [[cell complex]] $K$, a $k$-[[the algebra of chains|chain]] is a "formal" [[linear combination]] of finitely many ori is represented by a $6 \times 3$ [[matrix of a linear operator|matrix]].
  8 KB (1,318 words) - 18:42, 27 August 2015
• Cochains on graphs
  ==The algebra of plumbing, continued== We introduce more algebra with the familiar metaphor:
  15 KB (2,523 words) - 18:08, 28 November 2015
• Calculus on graphs
  ==The algebra of plumbing, continued== We introduce more algebra with the familiar metaphor.
  16 KB (2,578 words) - 00:14, 18 February 2016
• Cochain complexes and cohomology
  ...ollowing is very similar to the continuous case (why? because the [[linear algebra]] is the same). ...[[linear operator]], which is its $4 \times 4$ [[Linear operators: part 2|matrix]].
  17 KB (2,592 words) - 14:38, 14 April 2013
• The algebra of oriented cubes
  With the algebra we have learned, we can easily conclude the following about these cycles:
  32 KB (5,480 words) - 02:23, 26 March 2016
• Maps of graphs
  We can now move on to algebra. '''Exercise.''' Find the matrix of the chain map of a graph map that shifts by one edge a graph of $n$ edge
  29 KB (5,042 words) - 17:57, 28 November 2015
• MATLAB
  ...gineers (I mean <em>engineers</em>, not computer programmers) to explore [[matrix]] models interactively, then save their work as scripts - you were never me ...e mathematical chops to rearrange your problem into something solvable via matrix transformations, you can probably write it quickly and elegantly in Matlab
  4 KB (548 words) - 14:55, 9 October 2010
• How to compute Betti numbers
  ...lls. The coefficients of these linear combinations form the columns of the matrix. As such it is the solution of the matrix equation:
  5 KB (890 words) - 14:47, 24 August 2014
• Chain maps
  We can now move on to algebra. These three identities immediately give us the three columns of the matrix of the linear operator $f_0:C_0(K)\to C_0(L)$, or $f_0:{\bf R}^3\to {\bf R}
  47 KB (8,115 words) - 16:19, 20 July 2016
• Elementary Linear Algebra -- Spring 2018
  '''MTH 329 – Elementary Linear Algebra.''' Systems of linear equations, matrices and determinants, vector spaces, *Prerequisites: excellent algebra skills, working knowledge of basic Euclidean geometry, good understanding o
  4 KB (573 words) - 13:07, 7 April 2018

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