Primeros años de universidadAlgebraPlanificado
Linear algebra
Systems of linear equations and Gaussian elimination, matrices, vector spaces, linear maps, bases and dimension, rank–nullity, determinants, eigenvalues and diagonalisation.
math.algebra.linear-algebra
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Se basa en
Requisitos previos obligatorios, hasta el fondo: 20
Aprenderás a
- Solve linear systems by Gaussian elimination and describe solution sets as translates of null spaces.
- Prove that all bases of a finite-dimensional vector space have the same size, and apply the rank–nullity theorem.
- Represent linear maps by matrices, compose them by matrix multiplication and change basis.
- Compute eigenvalues and eigenvectors and decide whether a matrix is diagonalisable.
Definición
- A vector space over a field k is a set with addition and scalar multiplication satisfying the vector-space axioms.
- A linear map T : V → W satisfies T(u + v) = T(u) + T(v) and T(cv) = cT(v).
- A basis is a linearly independent spanning set; dim V is the number of elements in a basis.
- λ is an eigenvalue of T if Tv = λv for some non-zero vector v.
Teorema
- Rank–nullity: dim V = dim ker T + dim im T for a linear map T : V → W with V finite-dimensional.
- A composite of linear maps is linear, and corresponds to the product of their matrices.
- A square matrix is invertible if and only if its determinant is non-zero, if and only if its columns are linearly independent.
- Eigenvectors belonging to distinct eigenvalues are linearly independent.
Errores frecuentes
- Thinking matrix multiplication is commutative.
- Assuming every square matrix is diagonalisable; the matrix with rows (0, 1) and (0, 0) is not.
- Believing a linear map must send lines through the origin to lines; it may collapse them to a point.
Conduce a
Descubrimientos opcionales
- Categories Vector spaces and linear maps form a category: composites of linear maps are linear and identity maps are linear.
- Representation theory of finite groups A representation of a group is a homomorphism into the group of invertible linear maps of a vector space.
- ← Linear equations and inequalities
- ← The coordinate plane
Itinerarios de aprendizaje
Lecturas complementarias
- Sheldon Axler, Linear Algebra Done Right (1996)
- Gilbert Strang, Introduction to Linear Algebra