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Math. 513. Fall 2003. Midterm Exam
Part I. True False
(20 pts) Circle the number which represents the true statement (no expanation needs
to be given, no partial credit is given).
1. The inverse of a bijective linear map of linear spaces is a linear map.
2. A system of
m
linear equations with
n
unknowns is always solvable if
n > m
.
3. The rank of the sum of two matrices of the same size is always larger than the rank of each matrix.
4. The rank of the product of two square matrices is less or equal than the rank of each matrix.
5. Two linear spaces of the same (ﬁnite) dimension are isomorphic if and only if there exists a surjective
linear map from one space to another.
6. A ﬁnitedimensional linear space has inﬁnitely many diﬀerent bases.
7. The intersection of two linear subspaces of a vector space is a linear subspace.
8. A linear space of dimension
>
1 over the ﬁeld of real numbers has inﬁnitely many subspaces.
9. For any two square matrices
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 Winter '09
 IgorDolgachev
 Math, Linear Algebra, Algebra

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