### Check list for Midterm II

• Linear spaces and subspaces:
• What is a subspace of a vector space?
• How do you check a subset of a linear space is a subspace?
• What is the dimension of a linear space?
• Find a basis of a given linear space and determine its dimension.
• Which vector spaces are called finite dimensional?
• 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.
• Linear transformations
• Determine when a transformation is linear.
• Find the matrix of a linear transformation with respect to a given basis.
• Find the kernel, image, rank and nullity of a linear transformation between linear spaces.
• Find the basis of the kernel and image of a linear transformation.
• Determine whether a linear transformation is an isomorphism.
• What is a relation between matrices of a linear map with respect to two different bases?
• Coordinates
• Find the change of coordinates matrix between two given basis
• Find the coordinates of a vector with respect to a basis
• Orthogonality
• Determine whether a basis is orthonormal.
• Find the orthogonal projection of a vector in a linear space V onto a subspace of V.
• Find the orhogonal complement of a subspace of a linear space.
• Compute the length of  a vector.
• Find the angle between two vectors.
• Perform the Gram-Schmidt process
• Find the QR factorization of a matrix.

As usual, going over the past homeworks provides the best preparation. For extra practice, the following exercises are useful:

3.4: 13, 21, 27, 39;
4.1: 2,7,10, 25, 27, 31;
4.2: 7, 23, 25, 53, 67;
4.3: 7, 21, 24,42;
5.1: 7, 15, 21, 26, 29

Topics  you need to know (this is list includes just the highlights)
• 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?
• 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 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 vectors are said to be orthogonal?