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Math Help - True or False

  1. #1
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    True or False

    A linear system of two equations in three unknowns can not have exactly one solution. True or False?
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  2. #2
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    Re: True or False

    Think like this, we can find the values of variables if the number of equations and number of variables is same.
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  3. #3
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    Re: True or False

    rather than give a "formal" answer i'll give an "informal one". one can think of the dimension of a vector space as quantifying its "size" in some sense. a system of two equations in 3 unknowns is equivalent to a 2x3 matrix, which takes your inputs (the 3 unknowns) and spits out two numbers (the two equations evaluated at the inputs).

    so we have a mapping from a three-dimensional space to a two-dimensional one. this can't happen without at least one dimension "collapsing" down to 0. the dimension(s) that collapses is some line in the three-dimension space that gets mapped to a single point (the origin of the two-dimensional space). this is a many-to-one operation, which is NOT uniquely reversible.

    all of this can be formalized by using the concepts of rank, linear independence and spanning set, but the "idea" is what i just said above.
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    Re: True or False

    Quote Originally Posted by Civy71 View Post
    A linear system of two equations in three unknowns can not have exactly one solution. True or False?
    True

    xa +yb +zc = b
    xa +yb = b-zc
    where z can be assigned arbitrarily

    a.b,c,d are the columns of the two linear equations. x,y,z are the unknowns
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  5. #5
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    Re: True or False

    The rank of augmented matrix A:B and A have to be same(=r say) for solution to exist, if r=n unique solution, r<n, infinitely many solutions . In case of 2 equations and 3 variables the rank of augmented matrix and matrix A can be at the most 2 wich is less than 3, so infinitely many solutions exist wich can be found out by putting one unknown equal to a parameter and finding other unknowns in terms of that parameter.
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  6. #6
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    Re: True or False

    ax + by + cz = d is only valid if d is a LC of a,b and c.

    rank is less transparent and more wordy (in my opinion).
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