Consider a linear map fiven by f(x)=Ax, where A is an matrix. Prove that the image of f, i.e. the set , is equal to the column space Col(A).
A problem in Chemistry: The artificial sweetener Aspartame has chemical formula .
(a) Is it possible to prepare Aspartame from the following ingredients with no byproducts: Sucrose ( ), glucose ( ), water ( ), nitric acid ( ) and nitrogen pentoxide ( )? [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 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 ?
(a) Is the set
linearly independent in ?
(b) Is the set linearly independent in the vector space of all functions ?
5. Prove that the set of all matrices is a vector space where addition is addition of matrices, and scalar multiplication by is given by multiplying every entry of a matrix by .