Applied Linear Algebra

  1. Systems of Equations 1: I can
    • identify whether or not a matrix is in Reduced Row Echelon Form,
    • use Gaussian Elimination to put a matrix into Reduced Row Echelon Form,
    • use the Reduced Row Echelon Form of an augmented matrix to describe the solution space to a system of linear equations using appropriate notation.
  2. Systems of Equations 2: I can
    • use Pivot Positions to determine whether a linear system is consistent,
    • determine whether the solution to a consistent linear system is unique.
  3. Vectors and Matrices 1: I can
    • add vectors,
    • scale vectors by a real number,
    • determine whether a vector can be expressed as a linear combination of a set of given vectors.
  4. Vectors and Matrices 2: I can
    • add matrices,
    • scale matrices by a real number,
    • perform matrix-vector multiplication,
    • perform matrix-matrix multiplication,
    • translate a system of equations into a matrix equation of the form Ax = b.
  5. Vectors and Matrices 3: I can
    • determine whether a vector is in the span of a given set of vectors,
    • describe the span of a set of vectors as a set,
    • use the span to determine whether a linear system is consistent.
  6. Vectors and Matrices 4: I can
    • determine whether a given set of vectors is linearly independent.
  7. Vectors and Matrices 5: I can
    • define a matrix transformation,
    • find the domain and range of a matrix transformation,
    • compose matrix transformations,
    • determine whether a given function is a matrix transformation.
  8. Invertibility and Bases 1: I can
    • use Gaussian Elimination to determine whether a matrix is invertible,
    • find the inverse of an invertible matrix.
  9. Invertibility and Bases 2: I can
    • find a basis for a space,
    • translate between the coordinate systems for different bases.
  10. Invertibility and Bases 3: I can
    • compute the determinant of a given matrix,
    • use the determinant to determine whether a matrix is invertible.
  11. Invertibility and Bases 4: I can
    • find a basis for the null space of a given matrix,
    • find a basis for the column space of a given matrix,
    • determine whether a given set satisfies the definition of a vector space.
  12. Eigentheory 1: I can
    • determine whether a vector is an eigenvector of a given matrix,
    • find the eigenvalue associated to an eigenvector.
  13. Eigentheory 2: I can
    • use the characteristic polynomial to find the eigenvalues of a given matrix,
    • find the algebraic and geometric multiplicities of eigenvalues,
    • find a basis for the eigenspace associated to an eigenvalue.
  14. Eigentheory 3: I can
    • determine whether a given matrix is diagonalizable,
    • diagonalize a diagonalizable matrix.
  15. Orthogonality 1: I can
    • compute the dot product of two vectors,
    • use the dot product to find the angle between two vectors,
    • determine whether two vectors are orthogonal.
  16. Orthogonality 2: I can
    • compute the transpose of a matrix,
    • use the transpose to find a basis for the orthogonal complement of a given vector space.
  17. Orthogonality 3: I can
    • perform Orthogonal Decomposition.
  18. Orthogonality 4: I can
    • use Gram-Schmidt Orthogonalization to find an orthogonal basis for a given vector space,
    • find an orthonormal basis from an orthogonal basis for a given vector space.