MTEL Mathematics (Secondary) (63)

This chapter covers Objective 0002, complex numbers and linear algebra. It treats complex numbers in rectangular, trigonometric, and polar form, De Moivre's Theorem and complex roots, the Fundamental Theorem of Algebra, matrix operations and determinants, matrices as transformations of the plane, and vectors used to model forces and velocities.

The imaginary unit i satisfies i^2 = -1. A complex number has the form z = a + bi, where a = Re(z) is the real part and b = Im(z) is the imaginary part. The real numbers are the complex numbers with b = 0. Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal.

Powers of i cycle with period 4: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. To find i^n, divide n by 4 and use the remainder. For example, i^43 = i^3 = -i because 43 = 4 · 10 + 3.

Addition and subtraction work componentwise. Multiplication uses the distributive property and i^2 = -1: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

The conjugate of z = a + bi is z̄ = a - bi. The product z · z̄ = a^2 + b^2 is a nonnegative real number. The modulus is |z| = sqrt(a^2 + b^2), the distance from z to the origin in the complex plane, so z · z̄ = |z|^2. Division is done by multiplying numerator and denominator by the conjugate of the denominator.

Worked Example: Write (3 + 4i)/(1 - 2i) in the form a + bi.

Multiply numerator and denominator by the conjugate of the denominator, 1 + 2i.

Numerator: (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i^2 = 3 + 10i - 8 = -5 + 10i.

Denominator: (1 - 2i)(1 + 2i) = 1 - 4i^2 = 1 + 4 = 5. Divide each part by 5: (-5 + 10i)/5 = -1 + 2i.

In the complex plane (Argand diagram), z = a + bi is the point (a, b), or equivalently the vector <a, b>. Addition of complex numbers is vector addition, which is why rectangular form is also called vector form.

Every nonzero complex number can be written in trigonometric (polar) form z = r(cos θ + i sin θ), often abbreviated r cis θ, where r = |z| and θ is an argument of z with tan θ = b/a, adjusted for the quadrant. Converting back uses a = r cos θ and b = r sin θ.

Forms of a Complex Number

Multiplication in trigonometric form multiplies the moduli and adds the arguments: (r_1 cis θ_1)(r_2 cis θ_2) = r_1 r_2 cis(θ_1 + θ_2). Division divides the moduli and subtracts the arguments. Geometrically, multiplying by i rotates a point 90° counterclockwise about the origin, and multiplying by r cis θ rotates by θ and scales by r.

Common Trap: For z = -1 - i, the calculator's arctan(b/a) = arctan(1) gives 45°, but the point is in Quadrant III, so the correct argument is 225° (or -135°). Always sketch the point before choosing θ.

De Moivre's Theorem states that for any integer n, [r(cos θ + i sin θ)]^n = r^n (cos nθ + i sin nθ). It follows from the multiplication rule by induction.

Every nonzero complex number has exactly n distinct nth roots. If z = r cis θ, its nth roots are r^(1/n) cis((θ + 360°k)/n) for k = 0, 1, ..., n - 1. On the complex plane they lie on a circle of radius r^(1/n), equally spaced 360°/n apart, forming the vertices of a regular n-gon.

Worked Example: Find the three cube roots of 8i in rectangular form.

The modulus of each root is 8^(1/3) = 2.

The Fundamental Theorem of Algebra states that every polynomial of degree n ≥ 1 with complex coefficients has at least one complex zero. Applying the theorem repeatedly with the Factor Theorem shows that such a polynomial has exactly n zeros counted with multiplicity, and factors completely into n linear factors over the complex numbers.

When the coefficients are real, nonreal zeros come in conjugate pairs: if a + bi is a zero, so is a - bi. Consequently a real polynomial of odd degree must have at least one real zero.

For a quadratic ax^2 + bx + c = 0 with real coefficients, the discriminant D = b^2 - 4ac decides the nature of the solutions: two distinct real solutions if D > 0, one repeated real solution if D = 0, and two nonreal conjugate solutions if D < 0. For example, x^2 - 6x + 13 = 0 has D = 36 - 52 = -16, so x = (6 ± 4i)/2 = 3 ± 2i. Chapter 2.1 develops quadratic and polynomial solving in more detail.

A matrix is a rectangular array of numbers; an m × n matrix has m rows and n columns. Matrices of the same size are added entrywise, and a scalar multiple multiplies every entry. The product AB is defined only when the number of columns of A equals the number of rows of B; the (i, j) entry of AB is the dot product of row i of A with column j of B. An m × n matrix times an n × p matrix is m × p.

Properties of Matrix Operations

Non-commutativity is a favorite test point. For A = [[1, 1], [0, 1]] and B = [[1, 0], [1, 1]], AB = [[2, 1], [1, 1]] while BA = [[1, 1], [1, 2]].

The determinant of a 2 × 2 matrix [[a, b], [c, d]] is ad - bc. A square matrix is invertible exactly when its determinant is nonzero, and in the 2 × 2 case the inverse is (1/(ad - bc)) [[d, -b], [-c, a]]: swap the diagonal entries, negate the off-diagonal entries, and divide by the determinant. For 3 × 3 matrices, compute the determinant by cofactor expansion along a row or by the diagonal (Sarrus) method. Also useful: det(AB) = det(A) det(B), and (AB)^(-1) = B^(-1) A^(-1).

A linear system can be written as AX = B, where A is the coefficient matrix, X is the column of unknowns, and B is the column of constants. If A is invertible, the unique solution is X = A^(-1)B. If det A = 0, the system has either no solution or infinitely many. Cramer's Rule gives the same answer: x = det(A_x)/det(A) and y = det(A_y)/det(A), where A_x and A_y replace the corresponding column with the constants.

Worked Example: Solve 3x + 2y = 7 and 5x + 4y = 11 using an inverse matrix.

A 2 × 2 matrix M acts on a point (x, y), written as a column vector, by matrix multiplication. The columns of M tell where the unit vectors go: the first column is the image of (1, 0) and the second column is the image of (0, 1). That one fact lets you build or recognize any transformation quickly.

Standard Transformation Matrices

The absolute value of the determinant is the area scale factor: a region of area S is mapped to a region of area |det M| · S. A negative determinant means orientation is reversed (as with a reflection). A determinant of zero means the plane is collapsed onto a line or a point, so the transformation is not invertible. Composition of transformations corresponds to matrix multiplication, with the first transformation applied written on the right: "first A, then B" is the matrix BA.

Worked Example: The unit square with vertices (0, 0), (1, 0), (0, 1), and (1, 1) is transformed by M = [[2, 1], [1, 3]]. Find the image vertices and the area of the image.

The image of (0, 0) is (0, 0).

The image of (1, 0) is the first column of M: (2, 1).

The image of (0, 1) is the second column of M: (1, 3).

The image of (1, 1) is the sum of the columns: (3, 4).

The image is a parallelogram. det M = 2 · 3 - 1 · 1 = 5.

The original area is 1, so the image area is |5| · 1 = 5.

Result: The image has vertices (0, 0), (2, 1), (3, 4), (1, 3) and area 5.

A vector has magnitude and direction. In the plane, v = <v_1, v_2>, with magnitude |v| = sqrt(v_1^2 + v_2^2). A vector with magnitude r and direction angle θ (measured from the positive x-axis) has components <r cos θ, r sin θ>. Vectors add componentwise (the tip-to-tail or parallelogram rule), and a scalar multiple kv stretches v by |k| and reverses it if k < 0. A unit vector in the direction of v is v/|v|.

The dot product is u · v = u_1 v_1 + u_2 v_2, a scalar. It also satisfies u · v = |u| |v| cos θ, where θ is the angle between the vectors. Therefore u and v are perpendicular exactly when u · v = 0. The component (scalar projection) of u along v is (u · v)/|v|, which is what physics calls the part of a force acting in a given direction. Work done by a constant force F moving an object along displacement d is W = F · d.

θ = arccos(-0.5077) ≈ 120.5°. The negative dot product confirms an obtuse angle.

Result: The angle is about 120.5°.

Velocity problems treat wind or current as a vector added to the craft's own velocity. A plane flying with airspeed 200 km/h heading due north in a wind blowing 40 km/h due east has ground velocity <40, 200>. Its ground speed is sqrt(40^2 + 200^2) = sqrt(41,600) ≈ 204.0 km/h, and its course is arctan(40/200) ≈ 11.3° east of north. Force problems work the same way: an object is in equilibrium when the vector sum of all forces is the zero vector.

Exam Tip: Set the calculator to degree mode for vector and polar-form problems unless the question uses radians. A wrong mode produces answers that look plausible but are wrong, and MTEL distractors are built from exactly that mistake.

Divide complex numbers by multiplying by the conjugate of the denominator; |z|^2 = z · z̄, and powers of i cycle with period 4.

In trigonometric form, multiply moduli and add arguments; De Moivre gives powers, and the n nth roots are equally spaced by 360°/n on a circle of radius r^(1/n).

A degree-n polynomial has exactly n complex zeros with multiplicity; real polynomials have nonreal zeros in conjugate pairs.

Matrix multiplication is associative but not commutative; a 2 × 2 matrix is invertible exactly when ad - bc ≠ 0, and the inverse follows the swap-negate-divide pattern.

The columns of a transformation matrix are the images of (1, 0) and (0, 1), and |det M| is the area scale factor.

Use the dot product for angles, perpendicularity, projections, and work; add velocity or force vectors componentwise.