Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V ,

what is the relationship between S and the basis of V ?

Bound state in potential. Where

V(x)=inf for x<0

V(x)=0 for 0 ≤ x ≤ L

V (x) = U for x > L

Write down the Schrödingerequation and solutions (wavesolutions) on general form for the three V(x).

1. Let A be an m x n matrix. Prove that Ax = b has at least one solution for any b if and only if A has linearly independent rows.

2. Let V be a vector space with dimension 3, and let V = span(u, v, w). Prove that u, v, w are linearly independent (in other words, you are being asked to show that u, v, w form a basis for V)

1. Match the Following:

   -A.B.C.D.E.F.G.H.I.J. Bias -A.B.C.D.E.F.G.H.I.J. Double Blind -A.B.C.D.E.F.G.H.I.J. Clinical Trial -A.B.C.D.E.F.G.H.I.J. Cohort -A.B.C.D.E.F.G.H.I.J. Confounding -A.B.C.D.E.F.G.H.I.J. Cross-sectional -A.B.C.D.E.F.G.H.I.J. Disease Incidence -A.B.C.D.E.F.G.H.I.J. Intention to treat -A.B.C.D.E.F.G.H.I.J. Matching -A.B.C.D.E.F.G.H.I.J. Per protocol

   A. Risk factor distortion B. Systematic uncertainty C. Number of new case of disease over time D. Creating similar comparison groups E. Subject randomly assigned to treatment groups F. Subject and researcher unaware of treatment status G. At a single point in time H. Followed all treatment/study procedures I. Do not have to complete treatment J. Subjects with common characteristics followed over time

For each question below (2- 12), describe one feasible, ethical approach to exploring the relationship
using a case-control study, a cohort study, or a randomized trial. Be specific about the study population,
the primary research question, how you would allocate patients to cases/cohorts/treatments, and what outcomes you would observe.

1. Whether a new cholesterol drug is better than the current standard statin prescribed to lower
cholesterol
2. The relationship between high-stress jobs and death from heart disease
3. The effect of nutritional deficiencies due to poverty on brain development in children
4. The relationship between a new caffeine alternative and alertness (compared to caffeine)

Investigators interested in the association between serum vitamin D levels and various birth outcomes recruited pregnant mothers to have a blood draw during the second trimester of their pregnancy. Mothers were followed until they gave birth. The health outcomes were compared among mothers with low, adequate, and high vitamin D levels.

A) Describe the study as to its particular sampling method.

B) Describe a risk measurement that you think would be most appropriate to compare the groups, and justify your choice.

Choose the most appropriate option

Detail, with very specific examples of each item, the meaning of the following concepts related to statistical data gathering: Be sure that the concept is first defined, IN YOUR OWN WORDS, and that you use a concrete example (including visual examples) to illustrate your understanding of the statistical concept: Treatment Group; Control Group; Summary Statistic; Numerical Variable; Discrete Variable; Continuous Variable; Ordinal Variable; Nominal Variable; Dependent Variable; Independent Variable; Population; Sample; Simple Random Sample; Stratified Sample;Representative Sample; Convenience Sample; Observational Study; Cohort; Placebo; Prospective Study; Retrospective Study; Cluster; Control Group; Mean; Median; Mode; Histogram; Outlier; Pie Chart;

Answer each one, with (brief) justification. Throughout, V denotes a vector space, and bold-face letters like u, v, etc denote vectors in V.

A1. Suppose there are three linearly independent vectors in V. Is V necessarily finite-dimensional?

A2. Suppose there are three vectors in V that span V. Is V necessarily finite-dimensional?

A3. What is the dimension of the vector space of all polynomials of degree 4 or less?

A4. Can an infinite-dimensional vector space contain a finite-dimensional subspace?

A5, In a finite-dimensional vector space, is every spanning set necessarily finite?

A6. Can R³ contain a four-dimensional subspace?