MATH 262, Review for Midterm Test 2
Test topics
- Image and kernel of a linear transformation. Linear span. (3.1)
- Linear relations, linear independence, redundant vectors (3.2)
- Basis of a subspace (3.2)
- The dimension of a subspace of Rn (3.3)
- Coordinates. Matrices of linear transformations (3.4)
- Similar matrices (3.4)
- Vector spaces. Subspaces. Examples (4.1)
- Linear transformations. Isomorphisms (4.2)
- Coordinates. Matrices of linear transformations (4.3; skip "Change of basis")
Objectives/Skills
- Know the definitions of vector space, subspace, linear span, independence, basis, dimension,
linear transformations, kernels and images and be able to use them in examples
- Find redundant vectors in a list of vector-columns
- Find the kernel and image given a matrix of a linear transformation
- Find a basis of the image and the kernel. Find the dimension.
- Find coordinates of a vector in a given basis
- Given a matrix of a linear transformation in the standard basis find the matrix in a new basis
- Work with other examples of vector spaces besides Rn: matrices,
polynomials, complex numbers, sequences, functions.
Key facts
(You need to know the proofs (A), or at least be able to explain the main ideas when asked (C).)
- Linear independence can be established by checking for redundant vectors (see 3.2 or,
better, lecture notes)
- The number of vectors in a basis (Facts 3.3.1, 3.3.2 and 3.3.4)
- Rank-nullity theorem (Fact 3.3.7, w/o proof)
- Formula x= S cx for the coordinates (Definition 3.4.1)
- How to find a basis of a subspace (Summary 4.1.6)
- Properties of isomorphisms (Fact 4.2.4)
- Definitions: vector space, subspace, linear span, linear independence, basis,
dimension, coordinates, linear transformation, image, kernel, rank, nullity, matrix in a basis,
isomorphism, similar matrices
- Vector spaces Pn, Rm x n, C∞ (notations).
Review questions
(At the end of each chapter)
Chapter 3: 1-28, 31, 32, 35, 37, 38, 41, 43, 48;
Chapter 4: 1-8, 12-16, 18-21, 23, 25-31, 34, 35, 41, 42, 49, 56.
Format of the test
Problems on the test will not be taken from the homework but they will
often be similar. You will need to provide
a complete solution for each problem, showing all steps. You may get
partial credit for solutions that are not completely correct as long
as you have the right ideas and explain them. The test will
be closed books/notes. A simple scientific calculator (example: TI-30XII)
is allowed, but it is not required to answer the questions. Graphing
calculators (example: TI-83, 84, 89) may not be used.
A number of questions on the test will be simple yes/no type questions,
which may be taken from the review exercises in the end of each chapter.