Check list for MAT 211  Linear Algebra  Final


Chapter You need to know... ("Def" means definition...)
1
  • Def of linear combination
  • Dot product
  • Length of and angles between vectors.
  • Compute the distance between a point and a line in R2.
  • Understand and use the parametric equations of lines, planes and subspaces in Rn.
2
  • Def of a linear equations
  • Find the solution  of a  system of linear equations
  • Find the row-echelon form of a matrix.
  • Apply elementary row operations to a matrix.
  • Def of linear independence of a set of vectors.
  • State linear independence in different ways.
  • Def of the span of a set of vectors
  • How many solutions can a linear system have? (Hint: There are three possibilities)
3
  • Compute the product of two matrices
  • Def of invertible matrix
  • Find the inverse of a matrix, if possible, applying two different methods (Gauss-Jordan elimination and using determinants, as done in Chapter 4)
  • Solve a linear system of equations using the inverse of a matrix
  • Def. of elementary matrix.
  • Def. of subspace
  • Def of basis of a subspace
  • Def of dimension of a subspace
  • Def of the row and column space of a matrix
  • Def of nullspace of a matrix
  • Def of rank of a matrix
  • Find the coordinates of a vectors with respect to a given basis.
  • Def of linear transformation and relation to matrices
  • Composition of linear transformation and relation product of matrices
  • Inverse of a linear transformation (when possible) and relation to inverse of the corresponding matrix.
  • Find the matrix of a rotation in R2
4
  • Compute the determinant of a matrix
  • Properties of the determinant.
  • Relation between the inverse of an invertible matrix and its determinant.
  • Find the characteristic polynomial of a square matrix.
  • Definition of eigenvalues, eigenvectors and eigenspaces.
  • Find eigenvalues, eigenvectors and eigenspaces of a given matrix.
  • Compute the power of a matrix using eigenvalues and eigenvectors.
  • Def. of diagonalizable matrix
  • For a diagonalizable matrix, find an invertible matrix P and a diagonal matrix D such that P-1AP=D
  • Def of geometric and algebraic multiplicity of an eigenvalue
  • Relation between geometric and algebraic multiplicity of an eigenvalue.
  • Find the matrix of a reflection about a line in R2.
  • Find the matrix of a reflection about a plane in R3.
5
  • Def of orthonormal set.
  • Def of orthogonal set.
  • Relation between orthogonality and linear intependence
  • Write a vector in a subspace W as a linear combination of elements of an orthonormal basis of W
  • Write a vector in a subspace W as a linear combination of elements of an orthogonal basis of W
  • Def of orthogonal matrix
  • Inverse of an orthogonal matrix
  • Eigenvalues of an orthogonal matrix.
  • Def. of orthogonal complement of a subspace
  • Dimension of the orthogonal complement of a subspace
  • Find the orthogonal complement of a subspace.
  • Compute the orthogonal projection of a vector onto a subspace
  • Decompose a vector into an element of a subspace W and one element of the orthogonal complement of W
  • Given a basis of a subspace W, compute an orthonormal basis applying the Gram-Schmidt process.
  • Given a symmetric matrix A, find an orthogonal matrix Q and a diagonal matrix D such that D=QTAQ