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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

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