Linear algebra studies vector spaces and linear maps. Its central connection is
$$Ax=b\quad\Longleftrightarrow\quad b\text{ is a linear combination of the columns of }A.$$
Use this graph as both a course and a reference. Begin with the study guide, conventions and references, and test understanding with practice problems and worked solutions.
Main branches
- Linear systems: elimination, consistency, free variables and affine solution sets.
- Matrices: operations, rank, inverses, determinants, eigenvalues and factorizations.
- Systems as matrix equations: move between equations, column combinations and matrix notation.
- Vector spaces: subspaces, bases, coordinates, maps, inner products and abstract examples.
- Applications: differential equations, Markov chains, coding, numerical approximation and economic models.
Connecting ideas
Rank–nullity counts independent outputs and lost directions. Change of basis explains why one map has many matrix representations. Orthogonal projections lead to least squares. Eigenvectors reveal invariant directions, while SVD applies even when a matrix is rectangular or not diagonalizable.
For computation, see choosing factorizations and numerical error. Exact algebra, sensitivity to data, and the stability of an algorithm answer different questions.
Unless explicitly stated otherwise, scalars lie in $\mathbb R$ or $\mathbb C$, vectors are columns, and complex inner products use $\langle x,y\rangle=x^*y$. Individual results state required square, rank, field or finite-dimensional hypotheses.
Parents
Children
- Linear systems of equationsA linear system is a finite list of equations in the same unknowns, written $Ax=b$ with $A\in\mathbb F^{m\times n}$....
- MatricesMatrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are...
- Linear systems and matricesThis section covers the fundamental relationship between systems of linear equations and their matrix...
- Vector spacesVector spaces are one of the foundational structures in linear algebra. A vector space is a set of objects called...
- ApplicationsLinear algebra has wide-ranging applications across mathematics, science, engineering, economics, and computer...
- Linear Algebra study guide, conventions and referencesThis graph covers an undergraduate linear algebra sequence, with abstract operator theory as a second pass. 1....
- Linear Algebra practice problems with worked solutionsTry each problem before expanding its solution. Check answers by substitution or multiplication. Solve $x+2y-z=3$ and...
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