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?