7/27/2019 Linear Algebra - Exercise 1
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(10B1NMA731) APPLIED LINEAR ALGEBRA
EXERCISE-1 Vector Spaces and Subspaces
1. Define a vector space Vover a fieldF. Show that in a vector
space
(a) 00 =a (b) 00 = (c) = )1(
2. Let (V, +) be the set of all real valued functions defined
over [a, b] and R be the field of real
numbers. Show that V(R) is a vector space.
3. Let (V, +)be the set of all nm matrices with elements
belonging to a field F. Show that V(F)is a vector space. (Note: It
is denoted by nmF )
4. Let (V, +) be the set of all polynomials with coefficients
from a field F. Show that V(F) is a
vector space.
5. Show that C(C), C(R), C (Q), R(R), R (Q) and Q (Q) are vector
spaces. (In general F(F) is a
vector space)
6. Let ,...3,2,1, =nCn be the set of all n-tuples of complex
numbers. Show that it is a vector space
overC. Similarly, )(RR n and )(QQ n are vector spaces.
7. Let V be the set of all pairs ( )yx, of real numbers, and let
F be the field of real numbers.Examine each of the following cases
to find whetherV is a vector space over the field of real
numbers.(a) ( ) ( ) ( ) ( ) ( )0,,;0,,, 212211 axyxaxxyxyx
=+=+
(b) ( ) ( ) ( ) ( ) ( )yaxayxayyxxyxyx ,,;,,, 21212211 =++=+
(c) ( ) ( ) ( ) ( ) ( )yaxayxayyxxyxyx 2221212211 ,,;,,,
=++=+(d) ( ) ( ) ( ) ( ) ( )ayaxyxaxyyxyxyx ,,;,,, 21212211
=++=+
8. Define a subspace of a vector space. Let Wbe a subset ofV.
Show that Wis a subspace ofVif
and only if it is closed with vector addition and scalar
multiplication.
9. Let )},,{(3 zyxR = . Which of the following are subspaces of
3R ?
(a) ( ){ }RzyzyW = ,,,,01 (b) ( ){ }RzxzxW = ,,,1,2 (c) ( ){
}0,,,3 =++= zyxzyxW
(d) ( ){ }2
4 ,,, xyzyxW==
(e) ( ){ }0,,,5=
xzyxW (f)( ){ }1,,,6 =++= zyxzyxW
10. Let nnRV = be the vector space of all nn real matrices overR
and Wbe the subset of allnn nonsingular matrices. Is Wa subspace
ofV?
11. Let nixi :1, = be a given set of points (knots) on the real
line such that nxxx
