The Linear Logo

Dr. Mark V. Sapir


Class 1

Exercises

Homework due Class 2.

1. Find the solution set of each of the following linear equations.

  1. 7x - 5y = 3
  2. 3x1 - 5x2 + 4x3 = 7

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Class 2

Exercises

Homework due Class 3.

1. The curve y = ax2 + bx + c shown in the figure passes through the points (x1,y1), (x2,y2), and (x3,y3). Show that the coefficients, a, b, c are a solution of the system of linear equations whose augmented matrix is

[ x12 x1 1 y1 ]
[ x22 x2 1 y2 ]
[ x32 x3 1 y3 ]

2. For which value(s) of the constand k does the following system of linear eqations have not solutions? Exactly on solution? Infinitely many solutions?

x - y = 3
2x - 2y = k

Solutions

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Class 3

Exercises

Homework due Class 4.

1. The figure below shows the graph of a cubic equation y = ax3 + bx2 + cx + d. Find the coefficients a, b, c and d.


Solutions

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Class 4

Exercises

Homework due Class 5.

1. Solve each of the following systems by Gauss-Jordan elimination.

		 3x1 + 2x2 -  x3 = -15
		 5x1 + 3x2 + 2x3 =   0
		 3x1 +  x2 + 3x3 =  11
		-6x1 - 4x2 + 2x3 =  30



		      10y - 4z +  w =  1
		  x +  4y -  z +  w =  2
		 3x +  2y +  z + 2w =  5
		-2x -  8y + 2z - 2w = -4
		  x -  6y + 3z      =  1

2. For which values of a will the following system have no solutions? Exactly one solution? Infinitely many solutions? 
		  x +  2y -       3z =  4
		 3x -   y +       5z =  2
		 4x +   y + (a2-14)z =  a+2

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