12. Check to see if the following set of functions is orthogonal on the interval [0,3] with respect to the integral inner product. Show all work. Recall that a set of functions is orthogonal if all pairs of functions are orthogonal. πα {cos (777), sin (7), 1}

1. Check to see if the matrix below is (a) symmetric (b) skew-symmetric (c) orthogonal You need to check all three cases. 2. Compute the eigenvalues and verify that they satisfy the required properties for such matrix. (For example, if the matrix is symmetric, then the eigenvalues satisfy certain property. Likewise, if a matrix is skew-symmetric or if it is orthogonal.) 0.28 0.96 -0.96 0.28

14. (a) Find all the eigenvalues and their algebraic and geometric multiplicities. (b) Find bases for the corresponding eigenspaces. (c) Is the matrix is diagonalizable? Explain. If it is, then find P and D. 5 2 -9 6

13. Check to see if the following set of functions is orthogonal on the interval [-3, 3] with respect to the integral inner product. Show all work. Recall that a set of functions is orthogonal if all pairs of functions are orthogonal. Пх COS 3 {cos (), sin().1} 3

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A semiconductor manufacturer produces devices used as central processing units in personal computers. The speed of the devices (in megahertz) is important because it determines the price that the manufacturer can charge or the devices. Consider the observations in Sample A were generated from a specific process, whereas the observations in Sample B were generated from a different process. According to the comparative boxplot as follows, which conclusion is INCORRECT? Boxplot of Device Speed in Sample A and B 800 780 Speed of Device (MHz) 760 740 720 700 680 660 640 620 600 Sample A Sample B The median values for both Sample A and Sample B are essentially the same. Sample B contains an outlier. In Sample B, the number of devices with speeds less than 660 MHz is approximately equal to the number of devices with speeds greater than 700 MHz. In Sample A, the number of devices with speeds greater than 680 MHz is approximately 1.5 times the number of devices with speeds less than 680 MHz.

The histogram below displays the power consumption (in watts) of various industrial machines during a trial period, categorized into power consumption intervals. The rectangle for the interval 1000-1500 watts is missing. What should its height be? (Two decimal places.) Histogram of Power Consumption of Various Machines Relative Frequency 0.30 0.25 0.20 0.25 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 0.00 500 1000 1500 2000 2500 3000 3500 4000 Power Consumption (Watts)

5. By referring to the discussion in Sec. 14 related to Fig. 19 there, find a domain in the z plane whose image under the transformation w = z² is the square domain in the w plane bounded by the lines u = 1, u = 2, v = 1, and v = = 2. (See Fig. 2, Appendix 2.)