Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet x
Other exercises
Let be a field and let and be -vector spaces. Let
be a linear map. Prove that the graph of the map is a subspace of the Cartesian product .
Let be a field and let be a -vector space. Prove that for the map
is linear.
How does the graph of a linear map
look like? How can you see in a sketch of the graph the kernel of the map?
Let be a field and let and be -vector spaces. Let be a system of generators for and let be a family of vectors in .
a) Prove that there is at most one linear map
such that for all .
b) Give an example of such a situation, where there is no linear mapping with for all .
Let be a field and let be a -matrix and a -matrix over . Prove the following relationships concerning the rank
Prove that equality on the left occurs if is invertible, and equality on the right occurs if is invertible. Give an example of non-invertible matrices and such that equality on the left and on the right occurs.
Prove that the series
converges with sum equal to .
Examine for each of the following subsets the concepts upper bound, lower bound, supremum, infimum, maximum and minimum.
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .
Explain why the factorial function is continuous.