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Math 4377
Review Topics for In-Class Exam 1
1 Linear equations
1.1 Systems of linear equations
Know how to solve a linear system over a field F by forming the augmented matrix and applying row operations.
1.2 Row reduction and echelon forms
Does the solution set to a linear system change under elementary row opera- tions? What are independent/free variables? How can we tell there are free variables by looking at the row-reduced echelon form?
1.3 Vector equations
How can we rewrite a linear system Ax = b in vector form? Can we solve the system if b can be written as a linear combination of the column vectors of A?
1.4 Solution sets
How do the solutions to an inhomogeneous system relate to the solutions of the corresponding homogeneous one?
1.5 Matrix inverse
Know how to compute the inverse of a matrix. What is the inverse of AB in terms of A−^1 and B−^1? What are the properties of the system Ax = b if A is invertible?
2 Vector spaces
What are the defining properties of a vector space? Know the difference between finite and infinite-dimensional vector spaces.
2.1 Subspaces
Know how to test whether a subset of a vector space is a subspace.
2.2 Spanning sets and linear independence
What is the span of a set of vectors? When are vectors linearly independent? Does the span of the row vectors of a matrix A change under elementary row operations? What about the span of column vectors? If Ax = 0 has non-trivial solutions, what can we say about the column vectors of A?
2.3 Bases and dimension
You can always extend a linearly independent set to a basis by adding appropri- ate vectors to it. What condition is needed when adding a vector to a linearly independent set to preserve linear independence? Compute the dimension of W 1 + W 2 for subspaces W 1 and W 2. How many vectors are needed for a spanning set, how many can a linearly independent set have?
2.4 Coordinates and change of coordinates
How do we compute the change of coordinates matrix? How do we compute coordinates of a vector with respect to a new basis?
3 Linear Transformations
Tell the difference between linear and non-linear transformations. Know how to compare two linear transformations efficiently by invoking a basis.
3.1 Kernel and range, nullity and rank
What is the sum of nullity and rank? Describe the properties of a linear trans- formation with zero rank or with zero nullity.
