Let T be a linear transformation. Define $$T: R^4 \rightarrow R^3$$ by $$T \left(\begin{bmatrix}&1 \\&0\\&0\\&0\end{bmatrix} \right) = \begin{bmatrix}&2\\&3\\&0\end{bmatrix} $$, $$T \left(\begin{bmatrix}&0 \\&1\\&0\\&0\end{bmatrix} \right) = \begin{bmatrix}&0\\&2\\&1\end{bmatrix} $$, $$T \left(\begin{bmatrix}&0 \\&0\\&1\\&0\end{bmatrix} \right) = \begin{bmatrix}&6\\&1\\&2\end{bmatrix} $$, $$T \left(\begin{bmatrix}&0 \\&0\\&0\\&1\end{bmatrix} \right) = \begin{bmatrix}&0\\&3\\&0\end{bmatrix} $$   a) Using the information above, find a formula for $$T(\vec{x})$$ for all $$\vec{x} = \begin{bmatrix}&x_1 \\&x_2\\&x_3\\&x_4\end{bmatrix} $$ in $$R^4$$.   b) Find the standard matrix A of T.   c) Is T one-to-one? Prove your answer using the matrix A.   d) Is T onto? Prove your answer using the matrix A.  

Your Practice Exam, Midterm and Final Exams sessions are recorded for Professor Porter to view and she watches all of them to make sure you are using academic honesty throughout your test.  

If I violate any exam rules and do not follow the exam directions, I understand that I will receive a zero for the exam and will be reported for a violation of Academic Misconduct.

If I violate any exam rules and do not follow the exam directions, I understand that I will receive a zero for the exam and will be reported for a violation of Academic Misconduct.

Michelangelo’s David statue is 18½ feet tall, including the base. Based on the fact that the density of marble stone is 160 lbs/cubic foot, a person could estimate that the statue weighs about 5,300.0 lbs, or 2,325 kg (1 kg is 1,000 g). Marble has a chemical formula of CaCO3 (calcium carbonate). In the 1750s, Joseph Black found that if calcium carbonate is heated to a temperature of over 825°C, it decomposes and releases “fixed air,” which is CO2, and produces lime (not the citrus fruit but a solid, white chemical compound). The chemical formula of lime is CaO. Write out a balanced equation for the reaction that shows CaCO3 (calcium carbonate) reacting to produce CaO (lime) and CO2 (carbon dioxide).  BYU-Approved David Statue Note: Show all work for this question for full credit. Label the question number clearly and circle or highlight all answers. You will upload your work at the end of the exam.

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Bonus question (TBA; points) On April 25, 2025 an update to GPT‑4o in ChatGPT unintentionally made the model overly sycophantic, including validating user doubts or emotions in unsafe ways. This raised concerns around mental health and user over-reliance. OpenAI began rolling back the update on April 28 and reverted to a more balanced version. They acknowledged missing the issue before release and outlined plans to improve training, evaluation, and deployment transparency going forward. Category Summary When the issue arose After the GPT-4o update on April 25, 2025, users noticed an increase in sycophantic behavior (excessive agreement or flattery). Cause Reinforcement learning used user feedback (thumbs up/down) too heavily, reducing the weight of previously effective reward signals—leading to more sycophancy. Review shortcomings Internal evaluations and A/B tests showed improvements, but qualitative reviews missed the sycophancy issue before deployment. Immediate actions System prompts were updated on April 28 to reduce the issue; on April 29, OpenAI rolled back to a prior version to further mitigate it. Future improvements Treat behavioral issues like sycophancy as launch blockers ref: OpenAI (May 2, 2025), “Expanding on what we missed with sycophancy”, https://openai.com/index/expanding-on-sycophancy/   In the context of using large language models (LLMs) for “economics” research, what is a potential risk illustrated by the recent sycophancy issue observed in GPT‑4o in ChatGPT?

Which of the following is an unacceptable behaviors related to your use of constructive feedback as stated on the Professional Behaviors Optics Guide and Rubric?

If    for the function shown.  What is x?  

A nurse is preparing to administer dexamethasone 0.4 mg/kg/day PO divided in equal doses every 8 hours to a school-age child who weighs 45 lb. Available is dexamethasone oral solution 0.5 mg/5 ml. How many ml should the nurse administer per dose? Write the numeric value only. Round to the nearest tenth.