ECE 647 Linear Systems Theory I
2015-2016
Homework1
Q1) U is the space of all continuous functions between the time instants [t0, t1]=[0, 1].
A subspace of U is V where V is the space of continuous polynomials of degree
smaller than or equal to 2 defined between the time instants [t0, t1]=[0, 1]. One of the
spanning vector set of V is Va = {v1, v2 , v3 } = {1,t + t 2 , t - t 2 }
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 vectors are polynomial functions with real
coefficients the conjugate of a vector is also equal to itself. Thus 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 Vb,
which are orthogonal to each other and have the ability to span the same subspace V.
Q2) V is the vector set including all vectors in R3 and F is the field of real numbers.
Hence (V,F)=(R3,R) constitute a vector space. Lets define four subspaces of (V,F)
over the same field F whose vector sets are defined as:
ì æ v ö
ü
ïï ç 1 ÷
ïï
v  V1 , V1 = ív = ç v2 ÷ | v1 = v2 , v3 = 0ý
÷÷
ï çç
ï
ïî è v3 ø
ïþ
ì æ v ö
ü
ïï ç 1 ÷
ïï
v  V2 , V2 = ív = ç v2 ÷ | v1 = -v2 , v3 = 0ý
÷÷
ï çç
ï
ïî è v3 ø
ïþ
ì æ v ö
ü
ïï ç 1 ÷
ïï
v  V3 , V3 = ív = ç v2 ÷ | v2 = v3, v1 = 0ý
÷÷
ï çç
ï
ïî è v3 ø
ïþ
a) Is it possible to say that (V,F) is the direct sum of the subspaces (V1,F),
(V2,F), (V3,F)? Explain
b) Is it possible to say that (V,F) is the direct orthogonal sum of the subspaces
(V1,F), (V2,F), (V3,F)? Explain
Q3) Let A be a linear transformation from vector set R4 to vector set R4 for the field
of real numbers R.
A
æ
ö
A : ç (R 4 , R)®(R 4, R)÷
è
ø
This linear transformation can be shown by the matrix equation:
é 1 1 1 1 ù
ê
ú
1 2 2 1 ú
ê
y=
x
ê 2 0 1 1 ú
ê 4 3 4 3 ú
ë
û
where x and y are vectors 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) Find the orthogonal projection of vector v1 over the plane spanned by the vectors
v2 and v3. The vectors are given as:
é 1 ù
é 0 ù
é 1 ù
ê
ú
ê
ú
ê
ú
1 ú
2
0
ê
ê
ú,
ê
ú
v1 =
, v2 =
, v3 =
ê 0 ú
ê 1 ú
ê 1 ú
ê 0 ú
ê 0 ú
ê 1 ú
ë
û
ë
û
ë
û