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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
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.
independent vectors in a vector space V over
a field K, then u 1+ u2, u3— u4, u4+ u1 are
also linearly independent.
Linear Algebra
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.
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.
Linear Algebra
Solve by Gaussian elimination method the following system of equations :
x+y+z+t=5
x-y+z+t=1
x + z + t = 3
Linear Algebra
Let T : R3 -> R3 be the linear operator
defined by T(x1, x2, x3) = (x1, x3, -2x2 - x3).
Let f(x) = - x³ + 2. Find the operator f(T).
defined by T(x1, x2, x3) = (x1, x3, -2x2 - x3).
Let f(x) = - x³ + 2. Find the operator f(T).
Linear Algebra
Suppose a1 = (1, 0, 1), a2 = (0, 1, -2) and
a3 = (-1, -1, 0) are vectors in R³ and
f : R³ -> R is a linear functional such that
f(a1) = 1, f(a2) = -1 and f(a3) = 3. If
a = (p,q,r) belongs to R3, find f(a).
a3 = (-1, -1, 0) are vectors in R³ and
f : R³ -> R is a linear functional such that
f(a1) = 1, f(a2) = -1 and f(a3) = 3. If
a = (p,q,r) belongs to R3, find f(a).
Linear Algebra
Let B = f(a1,a2, a3) be an ordered basis of
R³ with a1 = (1, 0, -1), a2 = (1, 1, 1),
a3 = (1, 0, 0). Write the vector v = (a, b, c) as
a linear combination of the basis vectors
from B.
R³ with a1 = (1, 0, -1), a2 = (1, 1, 1),
a3 = (1, 0, 0). Write the vector v = (a, b, c) as
a linear combination of the basis vectors
from B.
Linear Algebra
company produces three products P, Q and R using raw materials A, B and C. One unit of P requires 1, 2 and 3 units of A, B and C respectively. One unit of Q requires 2, 3 and 2 units of A, B and C respectively. One unit of R requires 1, 2 and 2 units of A, B and C respectively. The number of units available for raw material A, B and C are 8, 14 and 13 units respectively. Using the matrix method, determine the number of units of each product to produce in order to utilize completely the available resources. (6 Marks)
Linear Algebra
Show whether the first three vectors are linearly independent
V1= 1, 1, 2, -2
V2= 2, -3, 0, 2
V3= -2, 0, 2, 2
V4= 3, -3 -2, 2
V1= 1, 1, 2, -2
V2= 2, -3, 0, 2
V3= -2, 0, 2, 2
V4= 3, -3 -2, 2
Linear Algebra
A company produces three products P, Q and R using raw materials A, B and C. One unit of P requires 1, 2 and 3 units of A, B and C respectively. One unit of Q requires 2, 3 and 2 units of A, B and C respectively. One unit of R requires 1, 2 and 2 units of A, B and C respectively. The number of units available for raw material A, B and C are 8, 14 and 13 units respectively. Using the matrix method, determine the number of units of each product to produce in order to utilize completely the available resources.
