Linear Algebra Exercises PDF

# Diagonalización de matrices - Although the power method only allows to estimate (if it exists) the strictly dominant eigenvalue of a matrix A, there are several variants of it that allow to estimate any of the eigenvalues of A from a good approximation A₀ of A. - For example, the invers...

# Diagonalización de matrices - Although the power method only allows to estimate (if it exists) the strictly dominant eigenvalue of a matrix A, there are several variants of it that allow to estimate any of the eigenvalues of A from a good approximation A₀ of A. - For example, the inverse power method # Ejercicios de Autoevaluación - Determine whether the following statements are true: 1. Two equivalent matrices are congruent if the matrices that relate them are transposed. 2. The P⁻¹AP operation can only be performed if A is square. 3. The vector ō cannot be an eigenvector. 4. If A is a regular matrix with an eigenvector, associated with an eigenvalue λ≠ 0, then v may not be an eigenvector of A⁻¹. 5. Eigenvectors corresponding to distinct eigenvalues can be linked. 6. If an eigenvalue, λ, of a matrix A has algebraic multiplicity 2, then the matrix must have 2 eigenvectors that are a basis of the subspace of eigenvectors associated with the eigenvalue λ. 7. It is true that the characteristic roots of two similar matrices are equal and, moreover, an endomorphism is diagonalizable if there is a basis of eigenvectors. 8. If A is a matrix of order n associated with an endomorphism f and λ is an eigenvalue of it, then the subspace L_λ of eigenvalue λ verifies: L_λ Nuc(f-λId) and Dim L_λ = n-rang(λI-A) being Id the linear identity application. 9. If A is a non-diagonalizable matrix of order 4 such that it has two distinct eigenvalues λ₁, λ₂, it may happen that d₁ = 2 and a₂ = 3. 10. If A is a matrix of order 8 and it only has an eigenvalue λ ≠ 1 with geometric dimension 7 then the Jordan matrix of A has only one one # Imagen de (2,3)_B en la base S': -1 -5 -12 3 -1 -23 # Ejercicios de Autoevaluación - Determine whether the following statements are true: 1. For an application f: V→W, with V and W vector spaces, to be linear it is sufficient that it verifies f(x+y)=f(x)+f(y) ∀x,y∈V. # CAPITULO 3/ Aplicaciones lineales y matrices - Determine whether the following statements are true: 1. If a system of vectors is free, the system that forms the images through a linear application will be free or linked depending on the application given. 2. The same matrix A can correspond to two linear applications f and g different for different bases. 3. A matrix A determines a linear application that does not depend on the bases chosen in the initial and final spaces. 4. The image subspace of a linear application ƒ has as dimension the rank of any associated matrix. 5. The kernel subspace Nuc(f) of a linear application f verifies that Dim Nuc(f) ≤ m - rang(A) being A any associated matrix of order n x m. 6. All linear applications are endomorphisms. 7. Base change matrices are singular. 8. Linear applications associated with a base change are the linear identity application. 9. Although the bases of the initial and final spaces of a linear application are changed, the matrix associated with the application does not vary. # CAPITULO 2/ Espacios vectoriales - As an immediate consequence of the definitions of base and direct sum we can affirm that a vector space is a direct sum of the subspaces generated by the vectors of any of its bases. - Theorem 2.9 If B = {v₁, v₂, ..., v_n} is a basis of the space V, then V = (v₁)⊕(v₂)⊕...⊕(v_n). # Ejercicios de Autoevaluación - Determine whether the following statements are true: 1. If in the set V = {(a, b) ∈ R²: b > 0} the internal operation (a, b)*(c, d) = (ac, b+d) is defined, then (V, *) has a neutral element. 2. If in a set A two different operations are defined: "+" and "*", the algebraic structures obtained in (A, +) and (A, *) coincide necessarily. 3. The set formed by the neutral element {ō} of a vector space V is always its subspace. 4. Although a subset U of a vector space V is also a vector space with different operations from those defined in the space V, we cannot say that U is a subspace of V. 5. The subset U = {(x₁, x₂) ∈ R²: x₁ = x₂} is not a commutative group. 6. The number of bases of a vector space depends on its dimension. 7. If a vector space is of dimension 3, it cannot have subspaces of dimension greater than 3. 8. The sum of two subspaces is always a direct sum. 9. The dimension of the sum of two subspaces is always the sum of their dimensions. 10. The sum of two subspaces is always contained in the union of them. # JLO 1/ Herramientas - Determine whether the following statements are true: 1. Each elementary row operation has a single associated elementary matrix. 2. In the definition of the determinant of a matrix of order n there is a sum of n! terms. 3. The inverse of a diagonal matrix, if it exists, is also diagonal. 4. If the determinant of a matrix of order n is zero then the rank is n. 5. A homogeneous system is compatible determined if and only if the rank of the coefficient matrix and the number of unknowns coincide. 6. If a system of linear equations has 5 equations, the rank of the extended matrix is 4 and it has 4 unknowns, the system has a unique solution. 7. If a system of linear equations has 5 equations, the rank of the coefficient matrix is 4 and it has 4 unknowns, the system can be incompatible. 8. Numerical methods applied to the resolution of compatible systems of equations always provide the exact solution. 9. The Gauss method consists of transforming the extended matrix into a reduced echelon form. 10. In the LU factorization method, operations of the type F→ AF; being A≠ 0 are allowed.