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
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
Read MoreKey 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
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
Read MoreSocial 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:
- Specify exactly what you want to find out.
- 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:
Read MoreProcess Costing and Standard Costing: Examples and Calculations
Process Costing System
Process Costing Example:
Company ”Alfa” produces product ”Y” through two production phases: assembly and testing. At the beginning of 200X, there was no work-in-progress balance. During 200X, they started production of 800 units of product ”Y”. Of these, 350 units were finished at the end of the period, and 450 units remained in the production process. All 450 unfinished units of product ”Y” have passed the assembly phase and are in the testing phase.
During 200X, the
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