Checklist for Midterm II

• What does the Kernel-Image (Rank-Nullity) 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 Rⁿ, 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, Cauchy-Schwarz inequality for vectors in an IPS?
• What is the Gram-Schmidt orthogonalization? Can you use it?