2) Let v, w, and x be vectors in Rn.
a) If v is the zero vector, what geometric object represents all linear
combinations of v?
b) Same question as a), except now for a nonzero v.
c) Same question as a) except now for nonzero vectors v and w (be care-
ful!).
d) Same question as a) except now for nonzero vectors v, w, and x (be
extra careful!).

Let V be the set of positive reals, V = {x ∈ R : x > 0}. Define “addition” on V by x“ + ”y = xy, and for α ∈ R, define “scalar multiplication” on V by “αx” = x^α . Is V a vector space with these unusual operations of addition and scalar multiplication? Prove your answer.

Suppose T ∈ L(V)

a) Show that V = ImT⁰ ⊃ ImT¹ ⊃ ImT² ⊃ ...

b) Show that if m ≥ 0 is such that ImT^m = ImT^{m + 1}, then ImT^{m + k} =ImT^{m + k + 1} for all k ≥ 0.

c) Show that if n = dim V, then ImTⁿ = ImT^{n + 1} = ImT^{n + 2} = ....

Let V and W be Banach spaces and suppose T : V → W is a linear map. Suppose that for every f ∈ W∗ the corresponding linear map f ◦ T on V is in V ∗ . Prove that T is bounded.

The potential of a particle is given as V (x) = -Vδ (x), with V being a positive number. Find the wave function and energy of the bounded states of this particle.

Let v and w be two nonzero vectors in R4 . Then v and w are linearly independent if only if v is not a scalar multiple of w. True or false?

DESIGN YOUR OWN EXPERIMENT

As a future scientist, you have the opportunity to show your laboratory skills by designing your own experiment. Below you will be provided an experimental question. Your job will be to physically complete an experiment to test the hypothesis associated with your question. Once again, you must conduct the actual experiment and document it. Use your Experimental Planning Form to guide you through the experiment. Keep in mind the overall goal is to conduct a valid experiment using the scientific method!!!!!

Topic: What is the effect of salt on the boiling temperature of water?

Below are more helpful pointers of details of how to document your experimental process.

Title

This should be short, but describe the experiment.

Problem Statement

What scientific question are you planning to investigate?

Background Information

Write a short paragraph describing some background information that you have learned about the topic of your investigation.

Hypothesis

You will propose a possible outcome for your experiment that includes the data that will support your hypothesis.

Materials List

Make a list of all of the materials you will use to complete your experiment.

Include consumable supplies and lab equipment in your list.

Procedure

This section must be written in a step-by-step numbered format, so that another student who is not in your group can follow it.

You can draw a picture to show how you will set-up your experiment (optional).

SAFETY – List what lab safety equipment you will need.

Data Table

Your data table should include a title, column headings, units (if needed) and a place to record all data from your experiment.

You must show at least three trials of data for your experiment, so be sure to leave room in your data table for all trails.

Conclusion Paragraph 1 –

Your conclusion should state if your data DID support your hypothesis or your data did NOT support your hypothesis and why (use data for support).

Summarize your data in a few sentences.

Paragraph 2 –

Explain what you learned about the problem statement from the experiment.

List some errors you and your partner(s) could have made while conducting the experiment that could affect your results. YOU MAY NOT SAY “NOTHING WENT WRONG” OR “THE EXPERIMENT WENT WELL”. Think here!!!!

The potential in a region between x = 0 and x = 6.00 m is V = a + bx, where a = 12.6 V and b = -7.90 V/m.

(a) Determine the potential at x = 0.
______ V

Determine the potential at x = 3.00 m.
______ V

Determine the potential at x = 6.00 m.
______ V

(b) Determine the magnitude and direction of the electric field at x = 0.

Determine the magnitude and direction of the electric field at x = 3.00 m.

Determine the magnitude and direction of the electric field at x = 6.00 m.

Problem 3. Throughout this problem, we fix a matrix A ∈ Fn,n with the property that A = A∗. (If F = R, then A is called symmetric. If F = C, then A is called Hermitian.) For u, v ∈ Fn,1, define [u, v] = v∗ Au. (a) Let Show that K is a subspace of Fn,1. K:={u∈Fn,1 :[u,v]=0forallv∈Fn,1}. (b) Suppose X is a subspace of Fn,1 with the property that [v,v] > 0 for all nonzero v ∈ X. (1) Show that [−, −] defines an inner product on X. (c) Suppose Y is a subspace of Fn,1 with the property that [v,v] < 0 for all nonzero v ∈ Y. (2) If X is a subspace with property (1), prove that X + K + Y is a direct sum, where K is defined in part (a).

The final product of the Calvin Cycle (the product that EXITS the Calvin Cycle, assuming successful carboxylation) is

To be in an active state, several enzymes of the Calvin Benson cycle require

The dry-weight mass of a plant (the weight of plant solids after water is removed) is from

Which statement is correct

The antennae complexes of photosynthesis gather light energy and channel it toward (choose ALL that apply)

Assuming linear electron flow from H₂O to NADP⁺
(i.e., the Z-scheme) what is the theoretical minimum number of photons required to produce one molecule of O₂ evolved (remember each water release a pair of electrons, it takes 2 H₂O to produce O₂)?

During CYCLIC electron flow, electrons from ferrodoxin are transferred to

What the heck am I talking about when I say “Reduction of plastoquinone to plastoquinol occurs on the stroma side of the thylakoid membrane, and oxidation of plastoquinol back to plastoquinone occurs on the lumen side of the thylakoid membrane”? This describes:

The Calvin-Benson cycle takes place predominantly in the

Put the following in the correct order for linear photosynthesis (the Z-scheme)

CF-type ATPase

Plastoquinone/plastiquinol

Photosystem II

Photosystem I

Cytochrome b6f

Plastocyanin