Mind-Warm-Ups Vector Spaces

Ultimate Space = S
si = arbitrary partition of S
Δsi = infinitesimal si, that is limit of si as it tends to zero.
pi = arbitrary point in S = Δsi

Problem 51: T is a one-to-one linear transformation from the m-dimensional linear space X into the n-dimensional linear space Y. What is the value of m if n2 = 9.
Ans: m = n = 3.

Problem 52: Identitfy the rigid motions that correspond to the following transformations:
(a) T(x, y) = (x+7, y-7).
(b) T1T2(x, y); where T1(x, y) = (x, -y) and T2(x,y) = (y, -x).
(c) T(x, y) = (xcosθ - ysinθ, xsinθ + ycosθ).
Ans: try it

Problem 53: What is the determinant of the rotation matrix of the following transformation:
T(x, y) = (xcosθ - ysinθ, xsinθ + ycosθ).
Ans: Rotation matrix A = ((aij)) matrix of order 2. Where a11 = cosθ, a12 = -sinθ, a21 = sinθ, a22 = cosθ
det(A) = a11a22 - a21a12 = cos2θ + sin2θ = 1.

Problem 54 : ((Aij)) and ((Bij)) are reflection matrices of reflections of the plane.
A11 = 0; A12 = 1; A21 = 1; A22= 0
B11 = 0; B12 = -1; B21 = -1; B22= 0
Determine:
(a) The mirror associated with ((Aij))
(b) The mirror associated with ((Bij))
Ans: (a) line y = x ; (b) y = -x

Problem 55 : A = ((aij)), is a scalar transformation matrix of order 2. a11 = k; a22 = 1/4.
Determine:
(a) if A represents a uniform stretching of the plane or a uniform compression of the plane.
(b) What change should be made in A inorder for it to represent a dilation of the plane.
(c) What is the difference between a transformation matrix representing a dilation of the plane and the transformation matrix representing a magnification of the plane.
Ans: (a) compression because k is 0< k <1
(b) Make k > 1
(c) Transformation matrix representing a magnification of the plane is a diagonal matrix of order 2 in which the diagonal elements are greater than zero and are not equal.

Problem 56: A = ((aij), is the matrix representing a a shear parallel to the x-axis.
a11 = 1; a12 = k; a21 = m; a22 = n.
A maps the vertices of a rectangle onto the vertices of a parallelogram as follows:
(0,0) -> (0,0); (2,0) -> (2,0);
(2,1) -> (5,1); (0,1) -> (3,1).
Determine the values k, m and n.
Ans: m and n are 0 and 1 respectively by definition. k = 3 (work out the transformation).

Problem 57: A = ((aij)) and B = ((bij)) are matrices representing projections of the plane.
a11 = 1; a12 = 0; a21 = 0; a22 = 0;
b11 = 1; b12 = 0; a21 = 1; a22 = 0.
Test for the nonsingularity of the transformations.
The point P(1,1) is projected under A, the result is then projected under B to the point P'.
Determine the coordinates of P'.
Ans: mappings that are not one-one are not nonsingular ( a matrice A is nonsingular if detA is not equal to zero. It is singular if detA =0). P'(1,1).

Problem 58: A = ((aij)) is an orthogonal matrix of order 2.
if a11 = 3/5; a12 = 4/5; a21 = -4/5; a22 = 3/5;
Determine the transpose and inverse of A
and whether A is proper or improper.
Ans: For an orthogonal matrix, AAT = I. The transpose and inverse of an orthogonal matrix are equal. proper if detA =1; improper if detA = -1.

Problem 59:A = ((aij)) is a translation matrix of order 3.
aii = 1 for all is; a12 = 0; a13 = x0;
a21 = 0; a23 = y0;
a31 = 0; a32 = 0.
Determine det A. What values of x0 and y0 will cause
every point on the plane to be a fixed point.
Ans: detA =1; x0 = 0; y0 =0.

Problem 60: A = ((aij)) is a square matrix of order 2.
if a11 = 3; a12 = 1; a21 = 2; a22 = 2;
Determine the eigenvalues of A.
What is the trace of A and its relationship to the eigenvalues.
Determine the eigenvectors associated with the eigenvalues of A.
Ans: form the characteristic function f(λ) = |A -λI|. Where I is the identity matrix. f(λ) = |A -λI| = 0; is the characteristic equation. The values of λ that satisfy the equation (its roots) are the eigenvalues of A.
So, |A -λI| = (3-λ)(2-λ) -2 = 0
Therefore, λ2 -5λ + 4 = (λ - 1)(λ - 4) = 0.
Therefore, λ1 =1 and λ2 = 4 are the roots of the characteristic equation and hence the eigenvalues of A.
Trace(A) = tr(A) = 3+2 =5 = sum of eigenvalues. Eigenvectors : any nonzero vector Xi such that (A -λiI)Xi = 0.

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Peter Oye Sagay

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