UNIT I VECTOR SPACES
1. Let R+ be the set of all positive real numbers. Define addition and scalar multiplication as
follows for all ; for all and . Determine whether or not R+ is a real vector space.
2. Let be the set of all matrices with real entries. Show that is a vector space over with
respect to usual matrix addition done entry wise and usual scalar multiplication done
entry wise. Verify all the conditions of a vector space.
3. Determine whether the set of all matrices of the form with the operations matrix addition
and scalar multiplication forms a vector space. If not list all the axioms that fail to told.
4. Determine whether the set of all matrices of the form with respect to standard matrix
addition and scalar multiplication is a vector space or not? If not, list all the axioms that
fail to hold.
5. Show that is a vector space over Q under usual operations.
6. Let W be the set of all points in satisfying the equation . Prove that W is the subspace
of .
7. Show that the set of all diagonal matrix is a subspace of .
8. Let . Prove that it is not a subspace of V.
9. Prove that the union of two subspaces of a vector space is a subspace if and only if one is
contained in the other.
10. If is a set of vectors in a vector space , prove that is the smallest subspace containing S.
11. Prove that is a linearly dependent set of vectors in V iff there exists a vector such that is
a linear combination of the proceeding vectors .
12. Prove that every linearly independent subset of a finitely generated vector space V(F) is
either a basis of V or can be extended to form a basis of V.
13. Let where and span R3.
14. Determine whether the set of vectors and . Show that S is a basis for R3.
15. is a vector space of polynomials of degree ≤ 3. Then express as a linear combinations of
and .
16. Determine whether or not the set forms a basis for .
17. Determine the basis and dimension of the solution space of the linear homogeneous
system .
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