Homework Answers
For the given A matrix and the condition AE = A, the E matrix is obtained by taking inverse of A and multiplying it with A. The E matrix will be equal to I matrix which will give us the condition that if a2 = 0 irrespective of the values of a1, the E matrix will exist.
The given function is the equation for the surface of cylinder. The cylinder is plotted.
The gradient is obtained at the given point (3,4,4) and then the equation of plane as the tangent is obtained.
The directional of z along the given unit vector is then obtained.
For the given value of θ, the X is obtained which turns out to be a identity matrix. Then AX can be obtained.
For the given values of A, B and C, the X vector is determined using algebraic methods.
(2) (a) For any O E [ 0 21] let -sino Cose x For Cosce sino...
(2) (a) For any O E [ 0 21] let -sino Cose x For Cosce sino 1² [ a b ] simplity any matrix A АХ 052 If A = and [33]... B =[2] C], find X-sored that A(x+B) = C. Q 2 (C) Let S be the set of matrices of the form As a a2 ag where arbitrary real numbers. Show there exists a unique matrix E in s such that A EA for all o in وگرنه...
O Q 2 (C) Let S be the set of matrices of the form A= a...
O Q 2 (C) Let S be the set of matrices of the form A= a az ag where a ja, are arbitrary real numbers. Show there exists a unique matrix such that A EA for all A in S.
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given ...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
3. Let V be the space of n X 1 matrices over C, with the inner...
3. Let V be the space of n X 1 matrices over C, with the inner product (X\Y) = YGX (where G is an n x n matrix such that this is an inner product). Let A be an n x n matrix and T the linear operator T(X) = AX. Find T*. If Y is a fixed element of V, find the element Z of V which determines the linear functional X + Y*X. In other words, find Z...
Let A, B, C and D be fixed n x n invertible matrices. Does the equation...
Let A, B, C and D be fixed n x n invertible matrices. Does the equation C(A - 2X)B =D have a solution for a n x n matrix X? If so, find it.
Let M be the set of 2 x 2 matrices of the form (82) where a,...
Let M be the set of 2 x 2 matrices of the form (82) where a, d ER-{0}. Consider the usual matrix multiplication, i.e: ae + bg af +bh ce + dg cf + dh (2)) = (ce ) (a) Show that (M,-) is an abelian group. (b) Compute the cyclic subgroup generated by M = What is the order of M? (6 -4) € M.
3. Let M2.2 be the vector space of 2 x 2 matrices. Let a) Find a...
3. Let M2.2 be the vector space of 2 x 2 matrices. Let a) Find a spanning set of U. b) What is the size of the smallest spanning set of U? c) Find a matrix A ? M2.2 such that A .
3. Let fRR' and g:R R2 be given by a) Write down the derivative matrices g'(u)...
3. Let fRR' and g:R R2 be given by a) Write down the derivative matrices g'(u) and f(r) and use the chain rule to find the derivative matrix (g fy (x) b) Are the entries of the new function(go f),(x) a linear or nonlinear function of z? mark 3 marks c) How do you understand the statement "(D() is a linear function" in Section 4.1 of the Class Notes?
4. Let M be the set of 2 x 2 matrices of the form (62) where...
4. Let M be the set of 2 x 2 matrices of the form (62) where a, d E R - {0}. Consider the usual matrix multiplication ·, i.e: ae + bg af + bh ce + dg cf + dh (a) Show that (M,·) is an abelian group. 1 (b) Compute the cyclic subgroup generated by M = What is the order of M? 66 -4) (1) EM EM.
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the ...
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the surface at the point P: (-2,1,3). Determine the equation of the line formed by the intersection of this plane with the plane z = 0. 10 marks (b) Find the directional derivative of the function F(r, y, z)at the point P: (1,-1,-2) in the direction of the vector Give a brief interpretation of what your result means. 2y -3 [9 marks]...
(2) (a) For any O E [ 0 21] let -sino Cose x For Cosce sino 1² [ a b ] simplity any matrix A АХ 052 If A = and [33]... B =[2] C], find X-sored that A(x+B) = C. Q 2 (C) Let S be the set of matrices of the form As a a2 ag where arbitrary real numbers. Show there exists a unique matrix E in s such that A EA for all o in وگرنه...
O Q 2 (C) Let S be the set of matrices of the form A= a az ag where a ja, are arbitrary real numbers. Show there exists a unique matrix such that A EA for all A in S.
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
3. Let V be the space of n X 1 matrices over C, with the inner product (X\Y) = YGX (where G is an n x n matrix such that this is an inner product). Let A be an n x n matrix and T the linear operator T(X) = AX. Find T*. If Y is a fixed element of V, find the element Z of V which determines the linear functional X + Y*X. In other words, find Z...
Let A, B, C and D be fixed n x n invertible matrices. Does the equation C(A - 2X)B =D have a solution for a n x n matrix X? If so, find it.
Let M be the set of 2 x 2 matrices of the form (82) where a, d ER-{0}. Consider the usual matrix multiplication, i.e: ae + bg af +bh ce + dg cf + dh (2)) = (ce ) (a) Show that (M,-) is an abelian group. (b) Compute the cyclic subgroup generated by M = What is the order of M? (6 -4) € M.
3. Let M2.2 be the vector space of 2 x 2 matrices. Let a) Find a spanning set of U. b) What is the size of the smallest spanning set of U? c) Find a matrix A ? M2.2 such that A .
3. Let fRR' and g:R R2 be given by a) Write down the derivative matrices g'(u) and f(r) and use the chain rule to find the derivative matrix (g fy (x) b) Are the entries of the new function(go f),(x) a linear or nonlinear function of z? mark 3 marks c) How do you understand the statement "(D() is a linear function" in Section 4.1 of the Class Notes?
4. Let M be the set of 2 x 2 matrices of the form (62) where a, d E R - {0}. Consider the usual matrix multiplication ·, i.e: ae + bg af + bh ce + dg cf + dh (a) Show that (M,·) is an abelian group. 1 (b) Compute the cyclic subgroup generated by M = What is the order of M? 66 -4) (1) EM EM.
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the surface at the point P: (-2,1,3). Determine the equation of the line formed by the intersection of this plane with the plane z = 0. 10 marks (b) Find the directional derivative of the function F(r, y, z)at the point P: (1,-1,-2) in the direction of the vector Give a brief interpretation of what your result means. 2y -3 [9 marks]...
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