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Linear Algebra - Two questions

  1. Sep 14, 2011 #1
    1. The problem statement, all variables and given/known data
    Q1: Find the value(s) of h for which the vectors are linearly dependent. Justify your answer.
    Q2: The linear transformation T is defined by T(x)=Ax. Find a vector x whose image under T is b and determine whether x is unique.


    2. Relevant equations
    Q1: I'm going to write the vectors as linear equations instead, since it should be easier to input:
    1(x1) -2(x2) + 3(x3)
    5(x1) -9(x2) + h(x3)
    -3(x1) +6(x2) -9(x3)

    Q2: I will write Matrix A with vector b in linear equation form as well.
    1(x1) -5(x2) -7(x3) = -2
    -3(x1) +7(x2) +5(x3) = -2


    3. The attempt at a solution
    I feel like I'm mostly right on these, but I would like some confirmation before I have to turn them in.

    For Q1: I noticed that row 3 is a scalar multiple of row 1, so I performed the necessary row operation to make it a zero row. Then I replaced row 2 with the sum of row 2 and (-5) row 1. This left me with:
    1(x1) - 2(x2) + 3(x3)
    0(x1) + 1(x2) +h-15(x3)
    0 0 0

    I'm a little fuzzy on linear dependency, but I thought that if I could make x3 a free variable, then I'd have it. But then I noticed that x3 is always going to be a free variable, so it seems to me that the solution should be all real numbers.

    For Q2 - I'll just say the row operations I did, and the solution that I came up with:
    (3)row 1 + row 2 replace row 2.
    Scale row 2 by -(1/8)
    (5)row 2 + row 1 replace row 1.

    This left me with:
    (x3) free
    (x1) = 3 - 3(x3)
    (x2) = 1 - 2(x3)
    And my answer is: Since (x3) is free, the solution is not unique.
    Thanks for any help.
     
    Last edited: Sep 14, 2011
  2. jcsd
  3. Sep 14, 2011 #2
    Take the determinant of the matrix and set it to 0
     
  4. Sep 14, 2011 #3

    Mark44

    Staff: Mentor

    These are NOT equations. If they were, each row would have an = in it.

    What are x1, x2, and x3? Are they vectors? Are they components of a single vector?

    Since you don't have a system of equations, what you're doing here doesn't seem valid to me.

    What is the exact wording of these problems?
     
  5. Sep 15, 2011 #4
    The exact wording of the problems is what I wrote.
    For Q1, just imagine a coeffecient matrix, now imagine them listed as column vectors V1-V3 instead.
     
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