513MT1 - Math 513 Fall 2003 Midterm Exam Part I True...

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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 finite-dimensional 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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