Essential Geometric Definitions and Theorems

Angles and Lines

  • Acute Angle: An angle that measures less than 90º.
  • Angles: Formed by two rays that share a common endpoint, provided that the two rays are non-collinear.
  • Angle Bisector: A ray that contains the vertex and divides the angle into two congruent angles.
  • Complementary Angle: A pair of angles that sum 90º.
  • Congruent Angles: Two angles are congruent if and only if they have equal measure.
  • Corresponding Angles: Angles that are created at the same location at each intersection where a transversal
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Understanding Vector Spaces, Linear Algebra, Eigenvalues, and Quadratic Forms

A.1.7. Vector Subspaces

A subset H of a vector space V is a vector subspace if and only if:

  • The zero vector 0 is in H.
  • For any vectors v1 and v2 in H, their sum v1 + v2 is also in H.
  • For any vector v1 in H and any scalar k, the scalar multiple k*v1 is also in H.

Examples of non-vector subspaces:

  • Sets defined by polynomial equations.
  • Vectors u = (x, y) where xy + x = 0.
  • Logarithmic functions.

Examples of vector subspaces:

  • Sets defined by linear equations like Ax + By + Cz = 0.
  • Sets defined by linear equations
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Understanding Floating-Point Representation and Errors

Floating-Point Representation and Errors

t: precision – a positive integer

β: base (or radix) – an integer ≥ 2 (2, 10, 16)

e: exponent – an integer

(decimal) value d1.d2d3 · · · dt × β -> (d1 + d2/β1 + · · · + dt/βt-1 ) × βe

exponent range emin ≤ e ≤ emax

1 + 2*((B -1)B(t-1) * (emax- emin + 1)) norm

1 + 2 * (Bt * (emax – emin + 1)) denorm

Memory stored in 3 fields: sign (1 bit positive negative), exponent (depends on range), fraction or significant (depends on precision)

1 + EpsM

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Key Marketing Research Variables and Editing

Scales of Measurement in Marketing Research

Understanding different scales is crucial for accurate data interpretation.

  • Nominal Scale: Numbers serve only as labels or tags for identifying and classifying objects. In marketing research, it’s used to identify respondents, brands, attributes, and other objects. Example: Numbers assigned to runners in a race.
  • Ordinal Scale: Indicates rank order, providing directional information in addition to nominal information. It measures non-numeric concepts like
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Statistics and Sampling: True or False Practice Questions

1) A population is a set that includes all elements about which we wish to draw a conclusion. True

2) If we examine some of the population measurements, we are conducting a census of the population. False. Example: A census is defined as examining all of the population measurements.

3) A random sample is selected so that every element in the population has the same chance of being included in the sample. True

4) An example of a quantitative variable is the manufacturer of a car. False

5) An example

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Social Research: Design, Objectives, and Approaches

Research Design

Coming to the planning of research to find out something scientific, you must design a strategy. For this, you must:

  1. Specify exactly what you want to find out.
  2. Decide how best to do it.

You have to make observations and interpret what has been observed. But before this, you must have a plan. You have to decide what is to be observed and analyzed: why and how. Herein is research design.

Research Objectives

Social research can have many goals, but three of the most common and useful are:

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