Homework #8  Consider the veracity or falsehood of each of the following statements. For bonus, argue for those that you believe are true while providing a counterexample for those that you believe are false.  If the first and third rows of A are equal, then det A 0.  If P is a projection, then uCP if and only if Pu  u.  If P is a projection, and detP  0, then P  I .  If A has determinant 10, then 1 A has determinant 1 10 .  If B is invertible, 1 1 det(A B ) det A (detB) .  If P is a projection, and R  2P I , then 2 R  I .  If P is a projection, and P  I , then detP  0 .  Short Computations. All of the following do not involve long computations:  Suppose 1 2 1 5 1 8 A                  and 1 9 2 4 3 1 A                   . Compute 7 13 19 A         .  Compute               0 8 7 1 0 2 3 4 5 3 0 9 2 0 0 0 3 0 0 0 1 9 3 2 0 det .  Use Cramer’s Rule to find 5 x (hint: you do not need your calculator). 1 2 3 4 5 5x 2x 8x x 3x 13 1 3 3x 5x 0 1 3 5 3x 3x 3x 9 1 2 3 5 3x 2x x 2x 7 1 3 x 4x 0 Let A 1 2 3 4 1 3 4 6 2 5 13 15 4 10 15 31 . Given is that det A  61. Do the following:  1 1 2 4 2 3 5 10 3 4 13 15 4 6 15 31 det  det2A  1 3 4 6 2 4 6 8 2 5 13 15 4 10 15 31 det  1 3 4 6 2 5 13 15 4 10 15 31 1 2 3 4 det  Consider the matrix A  0 1 0 0 0 0 1 0 0 0 0 1 1 2 2 1           . Use row (or column) expansion to compute det(xI A) .  The matrix 4 1 1 2 1 1 1 4 1 1 2 1 1 1 4 1 1 2 2 1 1 4 1 1 1 2 1 1 4 1 1 1 2 1 1 4 1 6 P is the projection matrix for the column space of matrix A. This matrix A is also known to be of full rank. Answer the following, giving reasons for your answers.  Find a transparent basis and the dimension for the column space of P.  Find a basis and the dimension for the column space of A .  What size is the matrix A ?  Find a transparent basis and the dimension for the null space of P.  Find a transparent basis and the dimension for the row space of P.  Find a basis and the dimension for the null space of A.  For which of the following b can you find a solution to the system Ax b ? This does not mean you should find a solution, only whether one could or not. 10 17 19 14 10 17 19 14 13 10 17 19 14 13 23 1 1 1 1 1 1 .  It is known that certain vector u is a solution to the system Ax c . Give all solutions to Ax c .  It is also known that 1 2 3 4 5 6 Ax does not have a solution. How would you change the constant vector so that there would be a solution? Extra Problems.  Fill in the blank with the best possible expression to complete the sentence truthfully. Only that one will be counted correct. 1. matrix with two equal columns will have zero determinant. 1 2 3 Some Every No 2. If A is invertible, then A commute with its inverse. 1 2 3 must always can will not 3. If A is 6  9 , then the columns of A be linearly independent. While in AT , the columns be linearly independent. 1 2 3 can have to cannot 4. Let A be square, and suppose Ax  0 has a nontrivial solution. Then detA equal 0. 1 2 3 may cannot must 5. Let A and B be 3 3. Then det (AB) equal det(A)det(B) . 1 2 3 could must couldn’t 6. Let A be square and suppose detA  0. Then have an inverse 1 2 3 will not may must always 7. Let A and B be 2  2 . Then det (A B) equal det(A)  det(B) . 1 2 3 could must could not 8. exist a 6  6 matrix all of whose entries are whole numbers and its determinant is 2 5 . 1 2 3 There does There does not There might Bonus: Consider the matrix 0 0 1 0 2 0 n 0 . Give its determinant as a function of n.

## Homework #8  Consider the veracity or falsehood of each of the following statements. For bonus, argue for those that you believe are true while providing a counterexample for those that you believe are false.  If the first and third rows of A are equal, then det A 0.  If P is a projection, then uCP if and only if Pu  u.  If P is a projection, and detP  0, then P  I .  If A has determinant 10, then 1 A has determinant 1 10 .  If B is invertible, 1 1 det(A B ) det A (detB) .  If P is a projection, and R  2P I , then 2 R  I .  If P is a projection, and P  I , then detP  0 .  Short Computations. All of the following do not involve long computations:  Suppose 1 2 1 5 1 8 A                  and 1 9 2 4 3 1 A                   . Compute 7 13 19 A         .  Compute               0 8 7 1 0 2 3 4 5 3 0 9 2 0 0 0 3 0 0 0 1 9 3 2 0 det .  Use Cramer’s Rule to find 5 x (hint: you do not need your calculator). 1 2 3 4 5 5x 2x 8x x 3x 13 1 3 3x 5x 0 1 3 5 3x 3x 3x 9 1 2 3 5 3x 2x x 2x 7 1 3 x 4x 0 Let A 1 2 3 4 1 3 4 6 2 5 13 15 4 10 15 31 . Given is that det A  61. Do the following:  1 1 2 4 2 3 5 10 3 4 13 15 4 6 15 31 det  det2A  1 3 4 6 2 4 6 8 2 5 13 15 4 10 15 31 det  1 3 4 6 2 5 13 15 4 10 15 31 1 2 3 4 det  Consider the matrix A  0 1 0 0 0 0 1 0 0 0 0 1 1 2 2 1           . Use row (or column) expansion to compute det(xI A) .  The matrix 4 1 1 2 1 1 1 4 1 1 2 1 1 1 4 1 1 2 2 1 1 4 1 1 1 2 1 1 4 1 1 1 2 1 1 4 1 6 P is the projection matrix for the column space of matrix A. This matrix A is also known to be of full rank. Answer the following, giving reasons for your answers.  Find a transparent basis and the dimension for the column space of P.  Find a basis and the dimension for the column space of A .  What size is the matrix A ?  Find a transparent basis and the dimension for the null space of P.  Find a transparent basis and the dimension for the row space of P.  Find a basis and the dimension for the null space of A.  For which of the following b can you find a solution to the system Ax b ? This does not mean you should find a solution, only whether one could or not. 10 17 19 14 10 17 19 14 13 10 17 19 14 13 23 1 1 1 1 1 1 .  It is known that certain vector u is a solution to the system Ax c . Give all solutions to Ax c .  It is also known that 1 2 3 4 5 6 Ax does not have a solution. How would you change the constant vector so that there would be a solution? Extra Problems.  Fill in the blank with the best possible expression to complete the sentence truthfully. Only that one will be counted correct. 1. matrix with two equal columns will have zero determinant. 1 2 3 Some Every No 2. If A is invertible, then A commute with its inverse. 1 2 3 must always can will not 3. If A is 6  9 , then the columns of A be linearly independent. While in AT , the columns be linearly independent. 1 2 3 can have to cannot 4. Let A be square, and suppose Ax  0 has a nontrivial solution. Then detA equal 0. 1 2 3 may cannot must 5. Let A and B be 3 3. Then det (AB) equal det(A)det(B) . 1 2 3 could must couldn’t 6. Let A be square and suppose detA  0. Then have an inverse 1 2 3 will not may must always 7. Let A and B be 2  2 . Then det (A B) equal det(A)  det(B) . 1 2 3 could must could not 8. exist a 6  6 matrix all of whose entries are whole numbers and its determinant is 2 5 . 1 2 3 There does There does not There might Bonus: Consider the matrix 0 0 1 0 2 0 n 0 . Give its determinant as a function of n.

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Name: Lab Time: BIO 218 Experiment Paper Rubric (20 points) General Formatting: (2 pts.) • Margins should be 1 inch top, bottom, left, and right. • Font should be 12 point Times New Roman or similar font. • Double-spaced. • Pages numbered. Title page is unnumbered. Next page is numbered at the bottom right corner with a 2 followed by pages 3, 4, and 5. • All sections must be included: Abstract, Introduction, Methods, Results, Discussion, and Literature Cited. • At least 3 pages (double spaced) but no more than five pages long. • All scientific names should be formatted correctly by italicizing and capitalizing the genus name and having the species name in lowercase (Bufo americanus). • Title page should have a specific title, student name, course, lab section time, and date. Project elements (18 pts. Total) • Abstract (2 points) o Summarize most important points using past tense. Use present tense to suggest a general conclusion which supports or refutes the hypothesis. • Introduction (3 points) o General background on topic and species (state scientific name!) o Discuss the possible tests of the hypothesis. o Reads from general to specific. o States hypothesis/hypotheses to be addressed. May discuss null and all alternative hypotheses. • Methods (2 points) o Reports how experiment was conducted and all materials used. Use enough detail so others could repeat the study. o Discuss the type(s) of data collected. o Discuss how data was to be analyzed/compared/used to test hypothesis. • Results (3 points) o Reports what happened in the experiment. o If comparisons made, discuss how they were made. o Report statistical and other data. Use “significant” only for statistical significance. o NO interpretation of data (no data analysis). o At least one original figure present and formatted correctly. Figures such as pictures and graphs are numbered and have captions underneath. o At least one table present and formatted correctly. Tables such as charts are numbered and have captions above them. • Discussion: (3 points) o Discusses the results of the experiment and ties in how the results fit with the literature. o Use past tense to discuss your results and shift to present tense to discuss previously published information. o States how results supported or refuted the original hypothesis. Hypotheses are never proven! o Ties in results with big picture within topic of biology. • Literature Cited: (2 points: .5 per citation) o At least 2 peer-reviewed journal articles (provided) + 2 peer-reviewed journal articles (found on your own). o References used in text properly. o References all listed in this section are alphabetized by author’s last name and formatted correctly. o All references listed in the Literature Cited section are cited in text. Writing Elements (3 pts.) • Grammar or spelling is error-free and excellent print quality. (1 pt) • Writing is clear and flows logically throughout paper. (1 pt) • Appropriate content in each section? (1 pt) Additional Comments:

## Name: Lab Time: BIO 218 Experiment Paper Rubric (20 points) General Formatting: (2 pts.) • Margins should be 1 inch top, bottom, left, and right. • Font should be 12 point Times New Roman or similar font. • Double-spaced. • Pages numbered. Title page is unnumbered. Next page is numbered at the bottom right corner with a 2 followed by pages 3, 4, and 5. • All sections must be included: Abstract, Introduction, Methods, Results, Discussion, and Literature Cited. • At least 3 pages (double spaced) but no more than five pages long. • All scientific names should be formatted correctly by italicizing and capitalizing the genus name and having the species name in lowercase (Bufo americanus). • Title page should have a specific title, student name, course, lab section time, and date. Project elements (18 pts. Total) • Abstract (2 points) o Summarize most important points using past tense. Use present tense to suggest a general conclusion which supports or refutes the hypothesis. • Introduction (3 points) o General background on topic and species (state scientific name!) o Discuss the possible tests of the hypothesis. o Reads from general to specific. o States hypothesis/hypotheses to be addressed. May discuss null and all alternative hypotheses. • Methods (2 points) o Reports how experiment was conducted and all materials used. Use enough detail so others could repeat the study. o Discuss the type(s) of data collected. o Discuss how data was to be analyzed/compared/used to test hypothesis. • Results (3 points) o Reports what happened in the experiment. o If comparisons made, discuss how they were made. o Report statistical and other data. Use “significant” only for statistical significance. o NO interpretation of data (no data analysis). o At least one original figure present and formatted correctly. Figures such as pictures and graphs are numbered and have captions underneath. o At least one table present and formatted correctly. Tables such as charts are numbered and have captions above them. • Discussion: (3 points) o Discusses the results of the experiment and ties in how the results fit with the literature. o Use past tense to discuss your results and shift to present tense to discuss previously published information. o States how results supported or refuted the original hypothesis. Hypotheses are never proven! o Ties in results with big picture within topic of biology. • Literature Cited: (2 points: .5 per citation) o At least 2 peer-reviewed journal articles (provided) + 2 peer-reviewed journal articles (found on your own). o References used in text properly. o References all listed in this section are alphabetized by author’s last name and formatted correctly. o All references listed in the Literature Cited section are cited in text. Writing Elements (3 pts.) • Grammar or spelling is error-free and excellent print quality. (1 pt) • Writing is clear and flows logically throughout paper. (1 pt) • Appropriate content in each section? (1 pt) Additional Comments:

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