MAT 342 Test 3 -- Review
4.1 Know the definition of a linear transformation on a vector space V. Understand the examples and the geometric meaning of the relevant examples. Be able to find the image of a subspace of V and the kernel of the linear transformation. Be able to prove that the ker(L) is a subspace of V.
4.2 Understand the matrix representation theorem. Be able to find the matrix that represents a linear transformation first with respect to the standard basis and then to any given ordered basis. (Graph on page 202 helps a lot.) Be able to determine the geometric meaning of (relevant) examples of linear transformations if the matrix representation is given.
5.1 Know the definitions of scalar product, length, distance, angle between two vectors in Rn. Be able to find scalar and vector projections of one vector onto another. Understand Cauchy-Schwartz inequality. Know when two vectors are orthogonal. Know and be able to prove the Pythagorian Law.
5.2 Know when two subspaces are orthogonal and what the orthogonal complement of a subspace is. Understand the Fundamental subspaces theorem that says N(A) = R(AT)┴ and
N(AT) = R(A)┴. Know that Rn is the direct sum of any subspace S and S┴.
5.3 Understand the idea of the least square solutions to an overdetermined system. Be able to set up and solve such problems.
5.4 Know the definition of an inner product on a vector space V Know the definitions of the norm (length), distance, angle between two vectors in inner product spaces. Be able to find scalar and vector projections of one vector onto another. Understand Cauchy-Schwartz inequality. Know when two vectors are orthogonal.
5.5 Know what orthonormal vectors, sets and basis are. Understand that orthonormal vectors are always linearly independent. Be able to prove that if a vector v in an inner product space V is the linear combination of orthonormal vectors (u1, u2 , … , un) then ci = v , ui >. Know what orthonormal matrices are and be familiar with their properties.
5.6 Know the Gram-Schmidt orthogonalization process or its modified version.
Problems recommended to practice on at this address: 13, 14, 15 16, 17, 42abc, 44, 45, 46, 47, 55, 57, 75, 76, 77, 78, 80, 85, 90, 92.