MA106Y18Tutorial1.pdf - Tutorial-1 MA 106(Linear Algebra Most of these problems are from reference texts for this course 1 Sketch the three lines and

MA106Y18Tutorial1.pdf - Tutorial-1 MA 106(Linear Algebra...

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Tutorial-1, MA 106 (Linear Algebra) Most of these problems are from reference texts for this course 1. Sketch the three lines, and decide if the system is solvable. If yes, find the solution set. x + 2 y = 2 , x - y = 2 , y = 1 2. For the equations x + y = 4, 2 x - 2 y = 4, draw the row picture (two intersecting lines) and the column picture (combination of two columns equal to the column vector (4 , 4) on the right side). 3. Describe the intersection of the three hyperplanes in a four dimensional space u + v + w + z = 6 , u + w + z = 4 , u + w = 2 Is it a line or a point or an empty set? What is the intersection if the fourth hyperplane u = - 1 is included? Find a different fourth equation that leaves us with no solution. 4. Under what conditions on y 1 , y 2 , y 3 , do the points (0 , y 1 ) , (1 , y 2 ) , (2 , y 3 ) lie on a line? 5. Starting with x + 4 y = 7, find the equation for the parallel line through x = 0 , y = 0. Find the equation of another line that meets the first at x = 3 , y = 1. 6. Starting with a first plane u + 2 v - w = 6, find the equation for (a) the parallel plane through the origin. (b) a second plane through origin that also contains the points (6 , 0 , 0) and (2 , 2 , 0). (c) a third plane that meets the first and second in the point (4 , 1 , 0). 7. It is impossible for a system of linear equations to have exactly two solutions. Explain why. (a) If ( x, y, z ) and ( X, Y, Z ) are two solutions of system of linear equations in 3 unknowns, what is another one? (b) If 25 planes meet at two points, where else do they meet? (How many variables are there in this system?) 8. Show that the set c 1 - 2 5 + c 2 - 5 2 | c 1 , c 2 R describes a line. Does it describe a line through origin? 9. Fill in the blanks. (a) For four linear equations in two unknowns x and y , the row picture shows four .
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  • Spring '16
  • Ravi Banavar

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