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 Rⁿ, 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 Kernel-Image (Rank-Nullity) 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, Cauchy-Schwarz inequality and thiangle inequality for vectors in an IPS?
• What is the Gram-Schidt orthogonalization?
• What is an orthogonal matrix?
• What is an orthogonal linear transformation?
• Can you say "A linear transformation is orthogonal" in 7 different ways?