exam2_sl - M341 Midterm Exam 2 56215 Solution I. (20...

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M341 Midterm Exam 2 56215 Solution I. (20 points) Mark the following statements either "True" or "False". Circle T for true and F for False. No proof or explanation is needed. 1. (T) If A , B are nonsingular n × n matrices, then (( AB ) T ) 1 = ( A 1 ) T ( B 1 ) T . 2. (F) The inverse of a type (I) row operation is a type (II) row operation. 3. (F) Span ( S ) is only de ned if S is a nite subset of a vector space. 4. (T) An n × n matrix A has determinant zero if and only if rank ( A ) < n . 5. (F) The set of all polynomial of degree 7 is a vector space under the usual operations of addition and scalar multiplication. 6. (F) Any plane W in R 3 is a subspace of R 3 under the usual operations. 7. (F) The set of all vectors of the form [0 , a, b, 1] is a subspace of R 4 under the usual operations. 8. (F) Let W 1 and W 2 be two subspaces of a vector space V . Then their union W 1 ∪ W 2 is also a subspace of V . 9. (T) If two rows of a square matrix A are identical, then | A | = 0 . 10. (T) If x is a linear combination of the rows of A , and B is row equivalent to A , then x is in the row space of B . II. Rank 1. (3 points) Give the de nition of the rank of a matrix A . Ans: The rank of a matrix A is the number of nonzero rows in the unique reduced row echelon form that is row equivalent to A . 2. (4 points) Let A and B be two matrices as following: A = 1 2 0 0 3 2 5 3 2 6 0 5 15 10 0 2 6 18 8 6 , B = 0 2 7 5 0 0 1 3 2 0 1 2 1 0 3 2 6 18 10 6 Find the rank of A and B .
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exam2_sl - M341 Midterm Exam 2 56215 Solution I. (20...

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