
Checklist for Midterm II
 What does the KernelImage (RankNullity) theorem say?
 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 8 axioms define a linear space (aka vector space)?
 Important examples of linear spaces: the space R^{n}, the space M_{nm} of all nxm matrices,
the space P_{n} of polynomials of degree n or less, the space of functions.
 What is a subspace of a linear 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 linear spaces are called finite dimensional?
 Which linear 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?
 What is an isomorphism?
 Which spaces are said to be isomorphic?
 What is a matrix of a linear transformation of a linear space? Can you find a matrix for a given basis?
 Which 4 axioms define an inner product space (IPS)?
 What is the Euclidean space?
 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 basis?
 What is the orthogonal complement of a subspace of an IPS?
 How to define the orthogonal projection onto a subspace?
 Can you state Pythagorean theorem, CauchySchwarz inequality for vectors in an IPS?
 What is the GramSchmidt orthogonalization? Can you use it?
