Advanced Linear Algebra (Math 424/524)

8 September 2002

These are just exercises to aid in reviewing your previous knowledge of linear algebra.

  1. Find a parametric description of the set of all solutions (x, y, z) of the system of linear equations

    x - 2 y + z = 0
    2 x - 3 y - z = 6
    		 .  

  2. Find the inverse of the matrix

     
    (
    1 -2
    1 -3
    		)
    		 .  

  3. Let M be the 3 \times 3 matrix that is given by

     M    =    
    (
    0 2 4
    1 0 1
    3 1 0
    		)
    		  .  
    Find the determinant of M.

  4. Let

     f:  R^{3}  ----->  R^{3} 
    be the map defined by f(x) = M x, where M is the matrix
     M    =    
    (
    0 2 4
    1 0 1
    -2 1 0
    		)
    		  .  

    1. What is the rank of the matrix M ?

    2. Find an equation for the image of f.

    3. Find a parametric representation of the fiber of f over the point (6, 2, -1).

    4. Find a point p in the image of f such that the vector drawn from the origin to p is perpendicular to the vector drawn from the origin to (6, 2, -1).

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