Sample problems for the final exam (3 pages!) Math 310
1. Consider bases U= [u1,u2] and V= [v1,v2] with the vectors
u1=µ1
3¶,u2=µ2
4¶,v1=µ2
5¶,and v2=µ1
3¶.
(a) What are the transition matrices Aand Bfor the basis change to the
standard basis [e1,e2] and from Uand Vrespectively?
(b) Use Aand Bto find the transition matrix Cfor the basis change from Uto
V.
2. Consider V= Span(1 + cos x−x, 1 + cos x, x), a subspace of C(−∞,∞).
(a) Find a basis of V:
(b) What is the dimension of V?
3. Consider the following chemical reaction: x1molecules of benzene burn with
the help of x2molecules of oxygen resulting in x3molecules of carbon dioxide
and x4molecules of water, i.e.
x1C6H6+x2O2→x3CO2+x4H2O
Balance the equation above by solving a system of linear equation.
4. Find the inverse of
A=
100
210
3−1 2
5. Given a system of linear equations
x+y=c1
x+ 2y=c2
x−y=c3
(a) reduce the system to row-echelon form. For what values does the system
have a solution?
(b) find the best approximate solution for c1= 1, c2= 2, c3= 1.
(c) find the best approximate solution for c1= 2, c2= 3, c3= 0.
6. The subspace Vof R3is spanned by
u1= (2,0,1)T,u2= (1,2,3)T,u3= (5,2,5)T.
Find a basis for its orthogonal complement V⊥.
7. Let [e1,e2,e3] be the standard basis of R3and
v1=
1
1
0
,v2=
1
0
1
,v3=
0
1
1
The linear operator L:R3→R3is defined by
L(v1) = v1, L(v2) = 2v2, L(v3) = 3v3.
1
