|Stony Brook University
||Introduction to Linear Algebra
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 Mnm of all nxm matrices, the space Pn 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?