Results 1 to 2 of 2

Thread: Few Urgent Linear Algebra Questions

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    76

    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; Nov 17th 2008 at 03:01 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    $\displaystyle n \in \mathbb{N}, I$ the $\displaystyle n\times n$ identity matrix, $\displaystyle A$ and $\displaystyle B$ two invertible $\displaystyle n\times n$ matrices.

    1)
    a) What about $\displaystyle I+(-I)$ ?

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

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

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

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

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


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

Similar Math Help Forum Discussions

  1. 2 Linear algebra questions
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Jan 7th 2012, 02:06 AM
  2. Linear algebra questions
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 19th 2009, 03:56 AM
  3. linear algebra urgent help please
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Aug 2nd 2008, 09:01 PM
  4. linear algebra questions
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Jun 28th 2008, 09:25 PM
  5. Urgent help needed with linear algebra problem
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Jun 13th 2005, 09:32 AM

Search Tags


/mathhelpforum @mathhelpforum