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Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

Conics

PDE

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

Conics

PDE

Ultimate Space = S

s_{i} = arbitrary partition of S

Δs_{i} = infinitesimal s_{i}, that is limit of s_{i} as it tends to zero.

p_{i} = arbitrary point in S = Δs_{i}

**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 n^{2} = 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) **T***1***T***2*(x, y); where **T***1*(x, y) = (x, -y) and **T***2*(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 = ((a_{ij})) matrix of order 2. Where a_{11} = cosθ, a_{12} = -sinθ, a_{21} = sinθ, a_{22} = cosθ
**

det(A) = a_{11}a_{22} - a_{21}a_{12} = cos^{2}θ + sin^{2}θ = 1.

**Problem 54 **: ((A_{ij})) and ((B_{ij})) are reflection matrices of reflections of the plane.

A_{11} = 0; A_{12} = 1; A_{21} = 1; A_{22}= 0

B_{11} = 0; B_{12} = -1; B_{21} = -1; B_{22}= 0

Determine:

(a) The *mirror* associated with ((A_{ij}))

(b) The *mirror* associated with ((B_{ij}))
**Ans: (a) line y = x ; (b) y = -x **

**Problem 55 **: **A** = ((a_{ij})), is a scalar transformation matrix of order 2. a_{11} = k; a_{22} = 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** = ((a_{ij}), is the matrix representing a *a shear parallel to the x-axis*.

a_{11} = 1; a_{12} = k; a_{21} = m; a_{22} = 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** = ((a_{ij})) and **B** = ((b_{ij})) are matrices representing **projections of the plane**.

a_{11} = 1; a_{12} = 0; a_{21} = 0; a_{22} = 0;

b_{11} = 1; b_{12} = 0; a_{21} = 1; a_{22} = 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** = ((a_{ij})) is an **orthogonal matrix** of order 2.

if a_{11} = 3/5; a_{12} = 4/5; a_{21} = -4/5; a_{22} = 3/5;

Determine the **transpose** and **inverse** of **A**

and whether **A** is **proper** or **improper**.
**Ans: For an orthogonal matrix, AA^{T} = I.
The transpose and inverse of an orthogonal matrix are equal. proper if detA =1; improper if detA = -1.
**

**Problem 59**:**A** = ((a_{ij})) is a **translation matrix** of order 3.

a_{ii} = 1 for all *i*s; a_{12} = 0; a_{13} = x_{0};

a_{21} = 0; a_{23} = y_{0};

a_{31} = 0; a_{32} = 0.

Determine **det A**. What values of x_{0} and y_{0} will cause

every point on the plane to be a *fixed point*.
**Ans: detA =1; x _{0} = 0; y_{0} =0. **

**Problem 60**: **A** = ((a_{ij})) is a square matrix of order 2.

if a_{11} = 3; a_{12} = 1; a_{21} = 2; a_{22} = 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, λ^{2} -5λ + 4 = (λ - 1)(λ - 4) = 0.

Therefore, λ_{1} =1 and λ_{2} = 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 X_{i} such that (A -λ_{i}I)X_{i} = 0.

Peter Oye Sagay