Perform arithmetic operations with complex numbers
Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Represent complex numbers and their operations on the complex plane
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the…
(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of…
(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of…
Use complex numbers in polynomial identities and equations
Solve quadratic equations with real coefficients that have complex solutions.
(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x - 2i).
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Represent and model with vector quantities.
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use…
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
Perform operations on vectors.
(+) Add and subtract vectors. There are 3 components within this standard
(+) Multiply a vector by a scalar. There are 2 components within this standard
Perform operations on matrices and use matrices in applications.
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
(+) Add, subtract, and multiply matrices of appropriate dimensions.
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still…
(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real…
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as…
(+) Work with 2 x 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.