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Linear Algebra
Find the general solutions of the systems whose augmented matrices is given?
[ 3 -4 2 0 ]
[ -9 12 -6 0 ]
[ -6 8 -4 0 ]
[ 3 -4 2 0 ]
[ -9 12 -6 0 ]
[ -6 8 -4 0 ]
Linear Algebra
A Leading mobile phone manufacturer is about to introduce a new series, initially they are launching 4 models of the same series. The accompanying table summarizes price and variable costs data, combined fixed costs equal $540,000.
Models
Infinity A Pro Max
Infinity A Pro
Infinity A
Infinity A Lite
Selling Price (in dollars)
500
400
340
220
Material Cost /unit (in dollars)
220
190
150
90
Labor Cost/unit (in dollars)
90
65
65
50
Table 1
a) Develop a joint total revenue function for sales of the four different models.
b) Develop an annual total cost function for manufacturing the four models.
c) Develop the profit function for sales of the four models.
d) Calculate the annual profit if the firm sells 9000, 12000, 45000 and 22000 units, respectively, of the four models?
Models
Infinity A Pro Max
Infinity A Pro
Infinity A
Infinity A Lite
Selling Price (in dollars)
500
400
340
220
Material Cost /unit (in dollars)
220
190
150
90
Labor Cost/unit (in dollars)
90
65
65
50
Table 1
a) Develop a joint total revenue function for sales of the four different models.
b) Develop an annual total cost function for manufacturing the four models.
c) Develop the profit function for sales of the four models.
d) Calculate the annual profit if the firm sells 9000, 12000, 45000 and 22000 units, respectively, of the four models?
Linear Algebra
Let T : R3 —> R3 be defined by
T(xi, x2, x3) = (3x1 + x3, - 2x1 + x2,
- x1 + 2x2 + 4x3)
Show that T^(-1) exists. Give the expression for T^(-1)(x1 , x2, x3) for T above.
Linear Algebra
There is no co-ordinate transformation that transforms the quadratic form x² + y²+ z² to xz + yz. True or false
Linear Algebra
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
1 0 0
0 -1/√2 1/√2
0 1/√2. a
]
is unitary? Justify your answer
Linear Algebra
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).
kernel of the linear transformation
T : R4 ->R4 defined by
T(x1, x2, x3, x4) = (x1 - x2, x2 - x3, x3 - x4, x4 - x1).
Linear Algebra
Reduce the quadratic form Q=x1²+2x2x3 to canonical form and hence find its nature, rank, index and signature.
Linear Algebra
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).
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).
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
Prove that C⁴/C ~_ C³
