**Igor Kriz**

**1.**

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*).

**2.**

*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).

**3.**

Which of the following sets are vector subspaces of ?

**4.**

(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 .

Tue Feb 10 12:12:32 EST 1998