Stony Brook University 

Introduction to Linear Algebra 
Mathematics Department 

MAT 211 
Julia Viro 

Spring 2009 
Check list for Midterm II
 What is the matrix of a change of a basis?
 How to find a matrix of a linear transformation with respect to the given bases in the domain and the target spaces?
 What is a relation between matrices of a linear map with respect to two different bases?
 Which matrices are called similar?
 Which 8 axioms define a vector space?
 Important examples of vector spaces: the coordinate space R^{n}, the space M_{nm} of all nxm matrices, the space P_{n} of all polynomials of degree less or equal n, the space of functions.
 What is a subspace of a vector space?
 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?
 Which vector spaces are called finite dimensional?
 Which vector spaces are called infinite dimensional?
 What is the dimension of a space?
 The dimension is the number of vectors in a basis.
 The dimension is the maximal number of linearly independent vectors.
 The dimension is the minimal number of spanning vectors.
 What are the coordinates of a vector with respect to a basis?
 What is a linear transformation?
 What is the kernel of a linear transformation?
 What is the image of a linear transformation?
 The rank of a linear transformation is the dimension of the image.
 What does the KernelImage (RankNullity) theorem say?
 What is an isomorphism?
 Which spaces are said to be isomorphic?
 Isomorphism is an equivalence relation.
 What is a matrix of a linear transformation?
 What is the change of basis matrix?
 Which 4 axioms define an inner product space (IPS)?
 What is the Euclidean space?
 What is the trace of a square matrix?
 What is the norm of a vector in IPS?
 What is the distance and the angle between two vectors in an IPS?
 Which vectors are said to be orthogonal?
 What is an orthonormal (ON) basis?
 What is the orthogonal compliment of a subspace of an IPS?
 How to define the orthogonal projection onto a subspace?
 Can you formulate Pythagoren theorem, CauchySchwarz inequality
and thiangle inequality for vectors in an IPS?
 What is the GramSchidt orthogonalization?
 What is an orthogonal matrix?
 What is an orthogonal linear transformation?
 Can you say "A linear transformation is orthogonal" in 7 different ways?