EXPLORING CONCEPTS
Method of Least SquaresFind a system of equations whose solution yields the coefficients
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Chapter 13 Solutions
Multivariable Calculus
- Writing Let x be a solution to mn homogeneous linear system of equations Ax=0. Explain why x is orthogonal to the row vectors of A.arrow_forwardElectrical Network Applying Kirchhoff’s Laws to the electrical network in the figure, the currents I1, I2, and I3, are the solution of the system I1I2+I3=03I1+2I2=72I2+4I3=0 Find the currents.arrow_forwardProof Let A be an mn matrix. a Prove that the system of linear equations Ax=b is consistent for all column vectors b if and only if the rank of A is m. b Prove that the homogeneous system of linear equations Ax=0 has only the trivial solution if and only if the columns of A are linearly independent.arrow_forward
- Adjusted Gross Income An individuals adjusted gross income is the amount of income that is subject to federal income tax. The following table shows the total adjusted gross income AGI, in trillions of dollars, reported to the IRS in the given year. Year 2009 2010 2011 2012 AGI trillions 7.6 8.1 8.6 9.1 a. Show that the data are linear. b. Let t denote the time in years since 2009, and let A denote the total adjusted gross income. Find a linear model for A as a function of t. c. Identify the slope of the linear model you found in part b, and explain its meaning in practical terms.arrow_forwardCellular Phone Subscribers The table shows the numbers of cellular phone subscribers y in millions in the United States from 2008 through 2013. Source: CTIA- The Wireless Association Year200820092010201120122013Number,y270286296316326336 (a) Find the least squares regression line for the data. Let x represent the year, with x=8 corresponding to 2008. (b) Use the linear regression capabilities of a graphing utility to find a linear model for the data. How does this model compare with the model obtained in part a? (c) Use the linear model to create a table of estimated values for y. Compare the estimated values with the actual data.arrow_forwardTemperature In Exercises 29 and 30, the figure shows the boundary temperatures in degree Celsius of an insulted thin metal plate. The steady-state temperature at an interior junction is approximately equal to the mean of the temperatures at the four surrounding junctions. Use a system of linear equations to approximate the interior temperatures T1,T2,T3, and T4.arrow_forward
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