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Math 296 problems 5

Igor Kriz


Consider a linear map tex2html_wrap_inline83 fiven by f(x)=Ax, where A is an tex2html_wrap_inline89 matrix. Prove that the image of f, i.e. the set tex2html_wrap_inline93 , is equal to the column space Col(A).


A problem in Chemistry: The artificial sweetener Aspartame has chemical formula tex2html_wrap_inline97 .

(a) Is it possible to prepare Aspartame from the following ingredients with no byproducts: Sucrose ( tex2html_wrap_inline99 ), glucose ( tex2html_wrap_inline101 ), water ( tex2html_wrap_inline103 ), nitric acid ( tex2html_wrap_inline105 ) and nitrogen pentoxide ( tex2html_wrap_inline107 )? [In chemistry, a formula consists of names of atoms in the molecule; the subscript stands for the number of atoms of an element in the molecule. H is hydrogen, C is carbon, O is oxygen and N is nitrogen. For example, a molecule of water has 2 atoms of hydrogen and one atom of oxygen.]

(b) The molecules in (a) can be represented by elements of tex2html_wrap_inline119 where the coordinates are quantities of atoms of the different elements C,H,O,N. Find a basis and dimension of the vector space V spanned by the vectors representing the molecules of the ingredients in (a).


Which of the following sets are vector subspaces of tex2html_wrap_inline125 ?



(a) Is the set


linearly independent in tex2html_wrap_inline119 ?

(b) Is the set tex2html_wrap_inline133 linearly independent in the vector space of all functions tex2html_wrap_inline135 ?

5. Prove that the set of all tex2html_wrap_inline89 matrices is a vector space where addition is addition of matrices, and scalar multiplication by tex2html_wrap_inline139 is given by multiplying every entry of a matrix by tex2html_wrap_inline141 .

Igor Kriz
Tue Feb 10 12:12:32 EST 1998