·Course Info

Checklist for Midterm I

  • How to solve a system of linear equations using Gauss-Jordan elimination?
  • What is the redused row-echelon form (rref) of a matrix, how to find it, and how the rref of a matrix gives the solution of a linear system?
  • What is the rank of a matrix?
  • How does the solution of a linear system depend on the ranks of coefficient- and augmented matrices?
  • How to add and multiply matrices?
  • Matrix multiplication is associative, but not commutative!
  • What is a vector? How to add vectors and take a scalar multiple of a vector? Which vectors are parallel?
  • What is Rn? Who lives there? What can we do with inhabitants?
  • How to calculate the dot product of two vectors in Rn.
  • What does it mean that two vectors are orthogonal?
  • What is a linear transformation? Can you give some examples?
  • What is the matrix of a linear transformation and how to find it?
  • Linear transformations on a plane: scaling, projection, reflection, rotation.
  • What is a composition of linear transformations and how to find its matrix?
  • What is the inverse thansformation?
  • What is the inverse matrix?
  • How to invert a matrix?
  • What is a subspace of Rn?
  • What is a linear combination of vectors?
  • What is a span of vectors?
  • Which vectors are said to be linearly dependent?
  • Which vectors are said to be linearly independent?
  • How to test linear dependence/independence?
  • What is the kernel of a linear transformation?
  • What is the image of a linear transformation?
  • What does it mean that vectors form a basis of a subspace?
  • What is the dimension of a subspace?
  • Can you say "A matrix is invertible" in nine different ways?
  • The rank of a matrix is the dimension of the image.
  • What are coordinates of a vector with respect to a basis?