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Linear Algebra
Let T(x1, x2, x3) = (x1+ x2, x2+ x3 , x1— x3) be
a linear operator on R³. Find its kernel. Show
that T is not onto. Show that (1, 1, 0) is in the
image of T. Also, find two distinct vectors
u1and u2such that T(u1 ) = (2, 2, 0) = T(u2).
a linear operator on R³. Find its kernel. Show
that T is not onto. Show that (1, 1, 0) is in the
image of T. Also, find two distinct vectors
u1and u2such that T(u1 ) = (2, 2, 0) = T(u2).
Linear Algebra
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).
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).
Linear Algebra
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
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
Linear Algebra
Reduce the quadratic form Q=3x2+5y2+3z2-2xy-2yz+2xz to canonical form and hencevfind its nature, rank, index and signature.
Linear Algebra
3. Use Cayley Hamilton theorem to find the value of the matrix given by
A8 − 5A7 + 7A6 − 3A5 + A4 − 5A3 + 8A2 − 2A + I if the matrix 𝐴 =
[
2 1 1
0 1 0
1 1 2
]
Linear Algebra
(a) Express cos7 θ in terms of multiples of angles.
(b) Express cos4 θ sin3 θ in terms of multiples of angles.
(c) Using complex numbers, prove that the, angles A, B and C of a planar triangle satisfy the relations (i) cos2 A + cos2 B + cos2 C = 1 − 2 cos A cos B cos C
(ii) sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.
Linear Algebra
Let u=(-1,0,2), v=(3,1,2), w=(1,-2,-2) be vectors in standard position. Compute
1. The orthogonal complement of v
1. The orthogonal complement of v
Linear Algebra
The coefficient matrix of any linear system must be square matrix.
True or false with correct explanation
True or false with correct explanation
Linear Algebra
Suppose A and B are 3x3 matrices. Show that If B is obtained from A by adding 2 times the first row of A to the last row of A, then det(A) =det(B).
Linear Algebra
Suppose A and B are 3x3 matrices. Prove that (i) (A+B)^T = B^T +A^T
(ii) det (AB)^T= det(A).det(B)
(ii) det (AB)^T= det(A).det(B)
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#340153 on Dec 2023
#340153 on Dec 2023