Page 72 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes
c
Where c 1 , ,c 3 are arbitrary scalars in F. Writing the set of all 5-tuples
2
x 1 ,x x x x 5
,
,
,
2
4
3
with x F such that
i
x 2x
2 1
x 3x 4x
4 1 2
It is clear that the vector 3, 6,1, 5,2 is in W, whereas 2,4,6,7,8 is not in W.
Self Assessment

1. Let V R 3 x , , : , ,z R and let W be the set of all triples x , ,z such that
y
y
z
y
x
x 3y 4z 0
Show that W is a subspace of V.
2. Prove that the set W of n-tuples x x 2 ,...x n 1 ,0 where all x’s belong to F, is a subspace of the
,
1
vector space V (F).
n
3. Show that the set W of the elements of the vector space V (R), of the form
3
y
y
x 2 , ,3y x , ,y R , is a subspace of V (R).
x
3
4. Let V be the space of all polynomial functions over F. Let S be the subset of V consisting
of the polynomial functions f , , f ,... defined by
f
0 1 2
f x x n , n 0,1,2,...
n
Show that W is the subspace spanned by the set S.
4
5. Show that the vector 3, 1,0, 1 is not in the subspace of R spanned by the vectors
2, 1,3,2 , 1,1,1, 3 and 1,1,9, 5 .

2.3 Summary

 If V is any vector space, V is a subspace of V; the subset consisting of the zero vector alone
is a space of V, and is called the zero subspace.
 Let F be a subfield of the field C of complex numbers, and let V be the vector space of all
2 × 2 matrices over F. Let W be the subset of V consisting of all matrices of the form
1
a b
c 0
 Consider a matrix A an m × n matrix over F and B and C are n × p matrices over F, then
A(a B + C) = a (AB) + AC
for each scalar a in F.

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