Stony Brook University 

Introduction to Linear Algebra 
Mathematics Department 

MAT 211 
Julia Viro 

Spring 2009 
Check list for Midterm I
 How to solve a system of linear equations using GaussJordan elimination?
 What is the redused rowechelon 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 accociative, 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 R^{n}? Who lives there? What can we do with
inhabitants?
 How to calculate the dot product of two vectors in R^{n}. What is the norn of a vector?
 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,
shear.
 What is a composition of linear transformations and how to find its matrix?
 What is the inverse thansformation?
 An isomorphism is an invertible linear transformation.
 What is the inverse matrix?
 How to invert a matrix?
 What is a subspace of R^{n}?
 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?
 What does the KernelImage (RankNullity) theorem say?
 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?
 How to find a matrix of a linear transformation with respect to some basis?
 What is a relation between matrices of a linear map with respect to two different bases?
 Which matrices are called similar?