Linear Algebra Answers

Are there values of (a) belongs to C for which the matrix [

1 0 0

0 -1/√2 1/√2

0 1/√2. a

]

is unitary? Justify your answer

Find the range space and a basis for the
kernel of the linear transformation
T : R4 ->R4 defined by
T(x1, x2, x3, x4) = (x1 - x2, x2 - x3, x3 - x4, x4 - x1).

Reduce the quadratic form Q=x1²+2x2x3 to canonical form and hence find its nature, rank, index and signature.

Which of the following statements are true and
which are false ? Give reasons for your
answers.
(a) Any square matrix with real entries is either symmetric or skew-symmetric or a linear combination of such matrices.
(b) If a linear operator has an eigenvalue 0,
then it cannot be one-one.
(c) If f : V --> K is a non-zero linear functional
and V a vector space of dimension n, then
there are n - 1 linearly independent vectors
v belongs to V such that f(v) = 0.
(d) Every binary operation on Rn is
commutative, for all n belongs to N.
(e) IAdj (A)I = IAI for all A belongs Mn(R).

Consider the real vector space
A = {(a, b, c, d) I a, b, c, d belongs to R,2a + 3b = c + d}.
Find dim A. Also find two distinct subspaces
B1. and B2 of R⁴ such that
A (direct sum) B1=R⁴=A(direct sum)B2

Prove that C⁴/C ~_ C³

Consider the real vector space Mn(R), of all
n x n matrices with entries from the set of
real numbers with respect to the usual
addition and scalar multiplication of
matrices. Find the smallest subspace of
Mn(R) which contains the identity matrix.
Also show that the set of all symmetric
matrices is a subspace of Mn(R).

Show that if u1 , u2, u3, u4 are linearly
independent vectors in a vector space V over
a field K, then u 1+ u2, u3— u4, u4+ u1 are
also linearly independent.

Let
W = ((X1, X2, X3) belongs to R³ : X2 + X3 = 0).
Show that W is a subspace of R³ . Find two
subspaces W1 and W2 of R3 such that
R³ = W (direct sum )W1 and R³ = W (direct sum) W2 but W1 (not equal to) W2.