21-06-2013
ECE 647
MIDTERM
Q1) (50 points). A set of vectors (functions) with real coefficients spanning a subspace of all
é
pù
continuous functions defined between the time instants éët0 t1 ùû = ê0
ú is
ë
2û
V = {v1, v2 , v3 } = {1, Sin(t),Cos(t)}.
The inner product of two vectors is defined as:
t1
x(t ), y (t )   x (t ) y (t )dt
t0
where x (t ) is the conjugate of x(t). As only the exponential functions with real coefficients
are available in the inner product, x (t )  x(t ) , and the inner product defined above can be
written also as,
t1
x(t ), y (t )   x(t ) y (t )dt
t0
Use Gram-Schmidt orthogonalization method to obtain a new set of vectors W which are
orthogonal to each other and have the ability to span the same subspace as V.
Q2) (20 points). Show that the set of complex functions V  e jnt n   N constitute a linearly
independent set of vectors.
N
Q3) (20 points). Let A be a linear transformation from vector set R5 to vector set R4 for the
field of real numbers R.
A


A :  ( R 5 , R ) ( R 4 , R ) 


This linear transformation can be shown by the matrix equation:
é 1 1 5 7 1 ù
ê
ú
3 1 9 15 0 ú
y =ê
x
ê 1 1 5 7 0 ú
ê 3 1 9 15 1 ú
ë
û
where x is a vector in R5 and y is a vector in R4.
a)
b)
c)
d)
Find a basis for the Range Space of linear transformation A (simply R(A))
Find a basis for the Null Space of linear transformation A (simply N(A))
Is this linear transformation 1 to 1? Explain your reasoning.
Is this linear transformation onto? Explain your reasoning.
Q4) (20 points) V is the vector set including all vectors in R4 and F is the field of real
numbers. Hence (V,F)=(R4,R) constitute a vector space. Lets define four subspaces of (V,F)
over the same field F whose vector sets are defined as:


 v1 
 




 v2 
v  V1 , V1  v    | v1  v 2 , v3  0, v 4  0
v


 3




 v4 




 v1 
 




 v2 
v  V2 , V2  v    | v 2  v3 , v1  0, v 4  0
v


 3
v 


 4




 v1 
 




 v2 
v  V3 , V3  v    | v3  v 4 , v1  0, v 2  0
v


 3




 v4 


ì æ v ö
ü
ï ç 1 ÷
ï
ï ç v2 ÷
ï
v  V4 , V4 = ív = ç
÷ | v2 = v4 , v1 = 0, v3 = 0ý
ï ç v3 ÷
ï
ï ç v ÷
ï
î è 4 ø
þ
a) Is it possible to say that (V,F) is the direct sum of the subspaces (V1,F), (V2,F), (V3,F)
and (V4,F)? Explain
b) Are the subspaces (V1,F) and (V4,F) orthogonal? Explain
c) Are the subspaces (V1,F) and (V3,F) orthogonal? Explain
d) Is it possible to say that (V,F) is the direct orthogonal sum of the subspaces (V1,F),
(V2,F), (V3,F) and (V4,F)? Explain