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Math Help - Linear systems

  1. #1
    Super Member Showcase_22's Avatar
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    Linear systems

    The linear system in x,y:

    (1+\lambda)x-\mu y=\delta

    (1-\lambda)x+ \mu y=2

    has a unique solution precisely when:

    a). \lambda=\pm \mu \delta
    b). \mu \neq \delta
    c). \lambda \mu =2\delta
    d). \mu \neq \delta
    What I did:

    \begin{pmatrix}<br />
{1+\lambda}&{-\mu}\\ <br />
{1-\lambda}&{\mu}<br />
\end{pmatrix}\begin{pmatrix}<br />
{x}\\ <br />
{y}<br />
\end{pmatrix}=\begin{pmatrix}<br />
{\delta}\\ <br />
{2}<br />
\end{pmatrix}

    The determinant of the matrix cannot be zero so:

    \mu+\mu \lambda+\mu -\mu \lambda=2 \mu

    Hence the answer is d).
    I have a nagging feling about that \delta though....

    Let T_1, T_2, T _3 : \Re^3 \rightarrow \Re ^2 be given by:

    T_1(x,y,z)=(x+1,y+z),
    T_2(x,y,z)=(2x,y)
    T_3(x,y,z)=(x^2,y+z)

    which of T_1,T_2 and T_3 is a linear transformation?

    a). T_1
    b). T_2
    c). T_3
    d). none of them
    I know that for a transformation to be linear:

    T(x+y)=T(x)+T(y)
    T(kx)=kT(x)

    applying these, T_2 is the only linear transformation so the answer is b).

    Can someone check these answers?
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  2. #2
    MHF Contributor

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    A matrix equation Ax= b has a unique solution if and only if det(A) is not 0. That has nothing to do with the right hand side and so nothing to do with \delta. The right hand side, and \delta, would be important if they asked whether there were no solutions or an infinite number of solutions.
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