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Math Help - Few Urgent Linear Algebra Questions

  1. #1
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    Few Urgent Linear Algebra Questions

    1. Determine which of the following formulas hold for all invertible nxn matrices A and B

    a. A+B is invertible
    b. (I - A)(I + A) = I - A^2
    c. A^9 is invertible
    d. (A + A^-1)^3 = A^3 + A^-3
    e. AB = BA
    f. (A+B)^2 = A^2 + B^2 + 2AB

    2. Determine the dimensions of the following subspaces of R^2 respectively R^3

    1. R^3= span{(-2, -12, 10), (-1, 3, -4), (2, 3, -1)}
    2. R^2= span{(–4, –2), (2, 0), (5, 0), (9, –1), (–2, 3)}
    3. R^2= span{(7, –2), (–6, 5)}
    4. R^2= span{(0, 0)}
    5. R^3= span{(–20, –4, 8), (–1, –4, –2), (10, 2, –4), (–4, 3, 4)}
    6. R^2= span{(2, –5, –7), (–6, 7, –8), (4, 8, –2)}
    Last edited by My Little Pony; November 17th 2008 at 04:01 AM.
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  2. #2
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    n \in \mathbb{N}, I the n\times n identity matrix, A and B two invertible n\times n matrices.

    1)
    a) What about I+(-I) ?

    b) (I-A)(I+A)=I^{2}+I.A-A.I-A^{2}=I+A-A-B^{2}=I-A^{2}.
    The first equality is true because M(n,K) is a ring.

    c) What about (A^{9})((A^{-1})^{9}) and associativity of multiplication.

    d) I^{-1}=I , so (I+I)^{3}=(2I)^{3}=8I\neq I^{3}+I^{-3}=2I

    e)Try with \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix} and \begin{pmatrix} 2&0 \\ 0&1 \end{pmatrix}

    f) What we can write in any ring is (A+B)^{2}=A^{2}+AB+BA+B^{2}
    You get the asked equality iff AB+BA=2AB, that is to say iff AB=BA. So e) answers the question.


    2) You just have to check how many vectors are independant in each set. Note that a subspace of \mathbb{R}^{n} has a maximal dimension of n, so if a set of vectors (x_{i})_{i\in I} contains at least n independant vectors, then span(x_{i}\ ;\ i\in I) = \mathbb{R}^{n}
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