| № |
Condition |
free/or 0.5$ |
| m96445 | Show that the inverse of a unitary matrix is again unitary. |
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| m96449 | Show that the linear system obtained by adding a multiple of an equation in (2) to another equation is equivalent to (2). |
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| m96450 | Show that the linear system obtained by interchanging two equations in (2) is equivalent to (2). |
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| m96457 | Show that the number of labeled trees with n vertices, k of which are pendant vertices, is (n - k)!S(n - 2, n - k) = (n!/k!)S(n - 2, n - k), where S(n - 2, n - k) is a Stirling number of the second kind. (This result was first established in 1959 by A. Renyi.) |
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| m96458 | Show that the number of partitions of a positive integer n where no summand appears more than twice equals the number of partitions of n where no summand is divisible by 3. |
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| m96459 | Show that the number of partitions of n ∈ Z+ where no summand is divisible by 4 equals the number of partitions of n where no even summand is repeated (although odd summands may or may not be repeated). |
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| m96460 | Show that the only subspaces of R1 are {0} and R1 itself. |
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| m96461 | Show that the only subspaces of R2 are {0}, R2, and any set consisting of all scalar multiples of a nonzero vector. |
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| m96462 | Show that the only vector x in R2 or R3 that is orthogonal to every other vector is the zero vector. |
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| m96467 | Show that the product of two 2 × 2 skew symmetric matrices is diagonal. Is this true for n × n skew symmetric matrices with n > 2? |
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| m96468 | Show that the product of two 2 × 2 skew symmetric matrices is diagonal. Is this true for n × n skew symmetric matrices with n > 2? |
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| m96493 | Show that u and v are parallel if and only if u x v = 0. |
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| m96497 | Show that when any edge is removed from K5, the resulting subgraph is planar. Is this true for the graph K3,3? |
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| m96510 | Simplify the following Boolean expressions.
(a) xy + (x + y) + y
(c) yz + wx + z + [wz(xy + wz)] |
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| m96511 | Simplify the following sum where n ∈ Z+:
(Hint: You may wish to start with the binomial theorem.) |
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| m96513 | Sixteen people are to be seated at two circular tables, one of which seats 10 while the other seats six. How many different seating arrangements are possible? |
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| m96570 | Sketch u and its image under each given matrix transformation f.
F: R2 → R2 defined by |
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| m96571 | Sketch u and its image under each given matrix transformation f.
F: R2 → R2 defined by |
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| m96576 | Solve each of die following for the real number x.
(a) (2 + xi)(3 - 2i) = 12 + 5i
(b) (2 + xi)(2 - xi) = 5 |
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| m96577 | Solve each of the following matrix equations for the 2 × 2 matrix A.
(a)
(b) |
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