Exam 1 Info
MCS 221 Exam 1 Info
Date & time: Monday, March 3
Place: Classroom
Coverage: Penney chapter 1, class notes
Test 1 will be a closed-book, closed-notes examination.
You may use one handwritten 3"-by-5" note card.
A copy of the handout on vector spaces and fields will be provided.
You may use a calculator for simple numerical calculations,
but not for entire matrix or linear system calculations.
There will not be any Maple problems on the test.
Be prepared to do problems and use knowledge from the following areas.
- 1.1 Vectors
- Ordered pairs, triples, n-tuples
- Vector addition and scalar multiplication
- Dot product
- Perpendicularity
- Length, distance
- Parametric representation of lines
- Proving equivalence of different line representations!
- Graphing points/vectors and lines
- 1.2 Matrices and vector space properties
- Matrix, M(m,n), row vector, column vector
- Matrix addition, scalar multiplication, transpose
- Linear combination, linear (in)dependence
- Span
- Vector space properties
- +Field properties
- Proving/disproving vector space properties!
- 1.3 Systems of linear equations
- Linear equation, linear system, solution (set)
- Equivalent systems, inconsistent equations
- Gaussian elimination!
- Parametric form of solution, translation vector,
spanning vectors!
- Rank
- Rn = M(n,1)
- 1.4 Gaussian elimination!
- Augmented matrix
- Elementary row operations
- Row equivalence (+an equivalence relation)
- Echelon form, reduced echelon form
- Back substitution
- Pivot variables, free variables
- More unknowns theorem.
- 1.5 Column space and null space
- Coefficient matrix, AX = B representation of a linear system
- Matrix product (m-by-n matrix times n-by-1 matrix)
- Linearity properties (of multiplication by a matrix)
- Column space
- Null space, homogeneous system
- Theorem 1 (on the solvability of AX=B)
- Translation theorem (characterizing the solutions of AX=B)
- The notion and characterization of subspaces
- Theorem 2: The null space is a subspace
The main computational technique to be tested is Gaussian elimination.
You should be able to carry out the row operations leading to echelon
or reduced echelon form, and you should be able to find the solutions
of linear systems and express them in parametric form.
You should be able to deal with two kinds of proof problems:
proof of the equivalence of two representations of a line
and proof that M(m,n) or ... satisfies specific vector space properties.
Of course, you should also be able to demonstrate your ability to carry
out routine vector and matrix calculations and your understanding of the
topics listed above.