Paper-V
Linear Algebra
Unit-I:
Vector spaces, General properties of vector spaces, Vector subspaces, Algebra of subspaces, linear
combination of vectors. Linear span, linear sum of two subspaces, Linear independence and dependence
of vectors, Basis of vector space, Finite dimensional vector spaces, Dimension of a vector space,
Dimension of a subspace.
Unit-II
Linear transformations, linear operators, Range and null space of a linear transformation, Rank and
nullity of a linear transformation, Rank - nullity Theorem, Invertible linear transformation.
Unit-III
The adjoint or transpose of a linear transformation, Sylvester’s law of nullity, characteristic values and
characteristic vectors , Cayley- Hamilton theorem.
Unit-IV
Inner product spaces, Euclidean and unitary spaces, Norm or length of a vector, Schwartz
inequality,Triangle inequality, Orthogonality, Orthonormal set, complete orthonormal set, Bessel’s
inequality, Gram - Schmidt orthogonalisation process.
PRESCRIBED TEXT BOOK:
Relevant Portions in A Text Book of B.Sc. MATHEMATICS Vol-III , V.Venkateswara Rao and
Krishna Murthy… S.Chand and Company Ltd.
Reference Book
Linear Algebra by J.N.Sharma and A.R.Vasista, Krishna Prakasham Mandir, Meerut-2
Sem-V
Paper-V
Max Marks : 75
Section – A
Answer any five questions
( 5 x 5 = 25 Marks )
1. If S is a subset of V(F) then prove that S is a subspace of V<=> L(S) = S
2. Show that the system of vectors (1,3,2) , (1,-7,-8) , (2,1,-1) of V3(R ) is linearly dependent.
3. The mapping T: V3(R )  V2(R ) is defined by T(x,y,z) = (x-y, x-z) .Show that T is a Linear
Transformation.
4. Find the nullspace, range, rank and nullity of a transformation T: V2(R )  V3(R) defined by
T(x, y) = ( x+y, x-y, y)
5. If A & B are similar linear Transformations on a vector space V .Prove that A* & B* are also
linear transformations on V.
6. Find the eigen values and eigen vectors of the matrix A =
5
1
4
2
7. Find a unit vector orthogonal to (4,2,3) in R3
8. If W is a subspace of the inner product space V(F) then prove that W┴ is also a subspace of V.
Section – B
Answer any five questions
( 5 x 10 = 50 Marks )
9. Let W1 and W2 be two subspaces of a finite dimensional vector space V(F) then prove that
dim(W1 + W2) = dimW1 + dimW2 –dim(W1∩W2)
10. Let W be a subspace of a finite dimensional vector space V(F) then prove that
dim(V/W) = dimV – dimW
11. State and prove that Rank - Nullity theorem.
12. Prove that the two finite dimensional vector spaces U & V over the same field F are
isomorphic iff they have the same dimension.
13. Find the characteristic roots and corresponding characteristic vectors of the matrix
8 −6 2
A = −6 7 −4
2 −4 3
2 1 2
14. If A =
5 3 3
1 0 −2
Verify Cayley Hamilton theorem hence find A-1
15. Prove that every finite dimensional inner product space has an orthonormal basis
16. Applying Gram-Schmidt process obtain an othonormal basis of R3 (R ) from the basis
{ (1,0,1),(1,0,-1),(0,3,4)}