Exam 3 Info
MCS 221 Exam 3 Info
UNDER CONSTRUCTION
Date & time: Friday, November 14, 2:30-3:20
Place: Classroom
Coverage: Linear transformations, inner products,
orthogonality, the Fundamental Theorem of Linear Algebra I & II,
finding bases for the four fundamental subspaces
Test 3 will be a closed-book, closed-notes examination.
You may use one new handwritten 3"-by-5" note card.
You may also use your Exam 1 and Exam 2 note cards.
You may use a calculator for simple numerical calculations,
but not for entire matrix or linear system calculations.
Topics probably covered:
- Elementary calculations: inner product, norm (length),
distance, angle or cosine thereof
- Orthogonal vectors, orthogonal matrices
- Orthogonal projection, Gram-Schmidt orthogonalization or orthonormalization
- Least-squares
- Fundamental Theorem of Linear Algebra
- Class notes: The Fundamental Theorem of Linear Algebra, Part I
Part I of this theorem tells us the dimensions of the
four fundamental subspaces of a matrix.
It incorporates these important theorems:
row rank = column rank (= rank);
rank+nullity = dimension.
Let A by an m-by-n matrix.
Let r = rank(A); r = # nonzero rows of any echelon form of A.
- Row space = span of rows of A; dimension = r.
- Null space = {x|Ax=0}; dimension = n-r.
- Column space = range of A; dimension = r.
- Left nullspace = nullspace of AT; dimension = m-r.
- Class notes: The Fundamental Theorem of Linear Algebra, Part II
Let A by an m-by-n real matrix.
- The null space of A is the orthogonal complement
of the row space of A, and vice versa.
- The left null space of A is the orthogonal
complement of the column space of A, and vice versa.
- Understand how the Fundamental Theorem describes the
action of a linear transformation.
- Be able to apply earlier matrix methods to find bases of
the four fundamental subspaces of a matrix.
An old exam from a galaxy far away