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The math 310 midterm 2 exam held on march 17, 2010. The exam covers topics such as vector addition, matrix operations, and finding bases and nullspaces. Students are required to determine the commutativity of vector addition, find the zero element in a given set, find the rowspace and nullspace of a matrix, and determine if certain sets are subspaces of r2ร2.
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โ, โ defined as:
u โ v = max(u, v)
k โ u = ku
(a) Determine whether vector addition is commutative; i.e. u โ v = v โ u for all u, v โ S.
(b) Determine whether or not S has a zero element 0 such that u โ 0 = u for all u โ S.
RREF โโ U =
(a) Find a basis for the rowspace of A.
(b) Find a basis for the nullspace of A.
(c) Are the columns of A linearly independent? If not, indicate any dependency relations amongst
them.
(d) Do the columns of A span R 4 ? Explain.
(e) What is the dimension of R(A)? Explain
โฅ , the orthogonal complement of S:
S = Span
(a) The set S 1 of all triangular 2 ร 2 matrices.
(b) The set S 2 of all symmetric 2 ร 2 matrices.
