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Math 296 problems 7

Igor Kriz

Regular problems:


Find the base change matrix from the basis


to the basis


[Use the definition.]


Recall that tex2html_wrap_inline69 denotes the vector space of polynomials of degrees tex2html_wrap_inline71 . Prove that the derivative is a linear map tex2html_wrap_inline73 . Find the matrix of this map with respect to the basis tex2html_wrap_inline75 in the source and the same basis in the target.


Prove that the identity tex2html_wrap_inline77 is a homeomorphism of metric spaces where the metric is the Euclidean metric in the source and the Postman metric tex2html_wrap_inline79 in the target.


Find a basis of the subspace of tex2html_wrap_inline81 consisting of all tex2html_wrap_inline83 such that


[It is a null space.]

Challenge problems:


By tex2html_wrap_inline87 we denote the vector space of all tex2html_wrap_inline89 matrices (row vectors). Now let A be an tex2html_wrap_inline93 matrix. The row space of A is the subspace of tex2html_wrap_inline87 spanned by the rows of A. Prove that the dimension of the column space of A (called the column rank) is equal to the dimension of the row space of A (called the row rank). [We already know that equivalent row operations preserve the dimension of the column space. Show that equivalent row operations actually preserve the row space. Thus, it suffices to verify the statement for matrices in reduced row echelon form.]


Prove that a map of metric spaces tex2html_wrap_inline105 is continuous if and only if for every open subset tex2html_wrap_inline107 , tex2html_wrap_inline109 is open in X ( tex2html_wrap_inline107 is open if for every tex2html_wrap_inline115 , there exists an tex2html_wrap_inline117 such that tex2html_wrap_inline119 ).

Igor Kriz
Sun Feb 22 23:31:30 EST 1998