Linear Algebra

If the company made a total of $301.50, how many children attended? Explain how you found your answer. 301.5-69-3.5S = 2.50

227. Prove that a determinant remains unchanged if from each row (except the last one) we subtract all succeeding rows.

3. Find the matrix of the linear transformation T: R² ->R² which rotates a vector by pi/4 radians counterclockwise.

Given the following picture: A cm B cm Find the table height (in centimetre) when A = 167 cm and B = 120 cm

Q2 On a Saturday night, at a party, you want to consume only vodka cocktails and gin cocktails. One vodka cocktail costs $8.75 and one gin cocktail costs $12.50. You want to taste both cocktails but you do not want to be too drunk so you decide to limit your total consumption of cocktails to 10 glasses. Your budget for this night is $110. Of course, you can only order integer values for glasses. Taking into account all the information above, how many different options do you have in total for the number of vodka and gin cocktails?

Given the following graph: find the vertices of the solutions set (in yellow in the graph). (Round your answers to 2 digits after the decimal point if needed) (sort your answers from the smallest value for x to the largest. If more than one coordinate has the same x value, then sort from the smallest value for y to the largest)

1. (S 5) Linear Functions A. Use g(x) = x + 2 to complete the following parts: 1. What is the y-intercept (write as a point)? Explain how to find the y-intercept algebraically. 2. What is the x-intercept (write as a point)? Explain how to find the x-intercept algebraically. 3. What is the domain? (use interval notation) 4. Is this linear function increasing or decreasing? Explain how you can tell just by looking at the equation. 5. Explain how to graph the function. You don't have to graph it (I know Desmos can do that for you!), I want you to explain how to graph it, using the values given in the function. B. If f(x) is a linear function, f(1) = -5 and f(5) = 6, find an equation for f(x). C. An electrician charges $25 for a service call plus $40 per hour of service. Write an equation in slope-intercept form for the cost, C, after h hours of service. What will be the total cost for 3 hours of work?

Problem 4. From the unit vector u (a) Show that Pu= u. Then u is an eigenvector with λ = 1. (b) If v is perpendicular to u show that Py=zero vector. Then λ = 0. (c) Find three independent eigenvectors of P all with eigenvalue λ = 0.

Problem 15. Decide on the stability or instability of dv/dt =w,dw/dt = v. Is there a solution that decays?

1. Two results that apply to scalars that students are tempted to apply to matrices are the following. • If ab = ac and a 0, then b = c(cancellation law). • ab = 0 if and only if a = 0 or b=0. These results do not extend to matrix arithmetic, in general. (a) Find matrices A, B and C such that AB = AC and A ‡ 02×2 but B‡ C. Justify your choices. (b) Find matrices A and B such that A ‡ 02x2 and B # 02x2 but AB = 02x2. Justify your choices. Recall that 02x2 represents 2 x 2 zero matrix.