Linear Algebra

Upon completion of MATH 2232 the student should be able to:

• solve systems of n equations in m unknowns using Gauss-Jordan elimination and Gaussian elimination
• prove and apply the basic properties of matrix addition, scalar multiplication, matrix multiplication, the transpose of a matrix and the inverse of a matrix
• express a system of equations as a matrix equation and vice versa
• determine the inverse of a matrix by Gauss-Jordan elimination and use the inverse to find the unique solution of a system of equations
• understand the terms square matrix, symmetric matrix, zero matrix, diagonal matrix, triangular matrix and identity matrix
• evaluate the determinant of an n x n matrix
• prove and apply the basic properties of the determinant of a matrix
• understand the terms singular, non-singular and invertible as applied to a matrix
• determine the adjoint of a matrix and use the adjoint to calculate the inverse of a matrix
• solve systems of equations using Cramer’s Rule
• prove, apply and explain the basic properties of vector addition and scalar multiplication on the vector space Rⁿ
• give the geometrical interpretation of subspaces of R² and R³
• prove that a given set of vectors is a subspace of R² or R³
• solve problems involving linear combinations, linear dependence, linear independence, the span of a set of vectors, bases and dimension in Rⁿ
• determine the rank of a matrix, the basis and dimension of the column space of a matrix and the basis and dimension of the row space of a matrix
• prove and apply the basic properties of the dot product and use the dot product to solve problems and define the norm of a vector, the angle between two vectors, the distance between two vectors and orthogonality in Rⁿ
• determine a basis for the set of vectors orthogonal to a given vector in Rⁿ
• calculate the projection of one vector onto another in Rⁿ
• explain the terms standard basis, orthogonal basis and orthonormal basis and be able to convert a basis into an orthonormal basis using the Gram-Schmidt Process (max of three vectors) in Rⁿ
• determine the various forms of the equations of lines and planes in three-space and be able to calculate the distance from a point to a plane and the distance from a point to a line
• prove that the set of polynomials of degree less than or equal to n, P_n, and the set of 2 x 2 matrices, M₂₂, are vector spaces
• determine which subset s of P₂ and M₂₂ are subspaces
• solve problems involving linear combinations, linear dependence, linear independence, the span of a set of vectors, basis and dimension in P₂ and M₂₂
• prove and apply the basic properties of an inner product in P₂ and M₂₂ and use the inner product to solve problems and define the norm of a vector, the angle between two vectors, the distance between two vectors and orthogonality
• prove or disprove that a given transformation is a linear transformation
• form composite transformations from given linear transformations
• determine the standard matrix for a linear transformation from Rⁿ to R^m
• determine the matrices that describe a rotation, a shear, a dilation or contraction and a reflection in R², and given a 2 x 2 matrix, describe the transformation in terms of the foregoing
• determine the kernel and range of a linear transformation and be able to express the solution as a basis of a subspace
• determine the rank and nullity of a linear transformation
• determine if a linear transformation is one-to-one
• determine the coordinate vectors of vectors in P₂ and M₂₂
• explain isomorphism of vector spaces
• find the transition matrix from one basis to another and the image of a given vector
• find the matrix of a linear transformation relative to given bases and the image of a given vector using the matrix of the transformation
• determine the characteristic polynomial, eigenvalues and corresponding eigenspaces of a given matrix
• prove that similar matrices have the same eigenvalues and use this property to diagonalise a square matrix
• compute the power of a square matrix using the fact that Aⁿ =PDⁿP^-1
• prove and apply the basic properties of the cross product and use the cross product to calculate the area of a triangle and the volume of a parallelepiped (optional)
• use the concept of orthogonal projection to find the least-squares solution of a system Ax=b (optional)
• use LU factorization to solve the linear system Ax=b (optional)
• prove the triangular inequality using the Cauchy-Schwartz Inequality (optional)
• solve systems of first order recurrence equations and second order recurrence (difference) equations (optional)
• apply techniques of linear algebra to solve problems related to: electrical network analysis, traffic flow, Fourier analysis, Leontif Input-Output models, Markov chains, and/or computer graphics (optional)