# UMBC MATH 430 - MATH 430 (Fall 2008) Exam 1 (7 pages)

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## MATH 430 (Fall 2008) Exam 1

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Lecture Notes

- Pages:
- 7
- School:
- University of Maryland, Baltimore County
- Course:
- Math 430 - Matrix Analysis

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NAME 1 15 2 6 3 16 4 10 5 10 6 8 7 10 T MATH 430 Fall 2008 Exam 1 Sep 29 No calculators books or notes Show all work and give complete explanations This 75 minute exam is worth a total of 75 points 1 15 pts a Define the nullspace and range of a matrix b State the Rank and Nullity Theorem and illustrate what it says in the context of a well chosen example 75 c Define the concept of a maximal linearly independent subset of a finite dimensional vector space B 2 6 pts Let A be a block matrix of the form A Prove that N A N B N C C 3 16 pts Let A be the matrix 1 2 3 4 9 7 A 3 7 4 8 18 14 Find bases for the four fundamental subspaces of A 4 10 pts a Prove that if a matrix is both symmetric and skew symmetric then it is zero b Without using matrices prove that the composition of two linear mappings between vector spaces is linear 5 10 pts a Let A be m n and B be n Prove that each column of AB can be expressed as a linear combination of the columns of A In particular find the coefficients in these linear combinations b Let 1 2 A 3 4 5 6 B 7 8 9 10 11 12 13 14 Use the formula you derived in a to calculate the 3rd column of AB 6 8 pts For each of the following statements either prove that the statement is true or give a specific counterexample a The union of two vector subspaces of a vector space is a vector subspace b The intersection of two vector subspaces of a vector space is a vector subspace 7 10 pts Find a basis for the vector space consisting of all 3 3 skew symmetric matrices and prove that it is indeed a basis Pledge I have neither given nor received aid on this exam Signature

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