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Math Help - [SOLVED] Bonus Question Linear Algebra

  1. #1
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    [SOLVED] Bonus Question Linear Algebra

    Hello people.

    I just had a midterm today and did really well. There was a bonus question on the test that I had no idea how to do. Could someone please show me?

    Prove or give a counter example:

    a) If S and T are subspaces of R^n, then S (union) T is also a subspace R^n
    is this b/c every element of s and every element of T are elements of R^n, so every element of S U T is also an element of R^n?
    b) If S and T are subspaces of R^n, then S (intersection) T is also a subspace of R^n
    is this b/c every element of s and every element of T are elements of R^n, so every element of S (intersection) T is also an element of R^n?


    lol, I guess this is a bonus bonus question, but my prof made this a try it yourself question in his notes and i have no clue how to do it....

    using the fact that rank(AB) <= rank(B) prove that if A and B are square matrices satisfying AB = I(sub)n then A^-1 = B and B^-1 = A

    all i know so far is that rank is the number of pivot columns of a matrix
    idk why our prof is introducing if then statements now, it's the last chapter in our textbook!
    anyway, help would be much appreciated!
    Last edited by Noxide; October 6th 2009 at 10:59 PM. Reason: thought of something else
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  2. #2
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    Quote Originally Posted by Noxide View Post
    Hello people.

    I just had a midterm today and did really well. There was a bonus question on the test that I had no idea how to do. Could someone please show me?

    Prove or give a counter example:

    a) If S and T are subspaces of R^n, then S (union) T is also a subspace R^n
    is this b/c every element of s and every element of T are elements of R^n, so every element of S U T is also an element of R^n?

    This is always false UNLESS one of them is contained in the other one.
    For example take S = {(x,y) in R^2; x = y} and T = {(x,0) in R^2}.
    Both S, T are subspaces but their union isn't since if you take (1,1) in S and (1,0) in T and you sum them you get (2,1), which isn't neither in S nor in T.

    b) If S and T are subspaces of R^n, then S (intersection) T is also a subspace of R^n
    is this b/c every element of s and every element of T are elements of R^n, so every element of S (intersection) T is also an element of R^n?


    I don't understand what you mean by "is this b/c every element of s and every element of T are elements of R^n, so every element of S (intersection) T is also an element of R^n?"...in fact, I douibt whether you have an accurate idea what this means.
    Anyway this is always true and you can easily prove it using the definition of subspace.

    Tonio

    lol, I guess this is a bonus bonus question, but my prof made this a try it yourself question in his notes and i have no clue how to do it....

    using the fact that rank(AB) <= rank(B) prove that if A and B are square matrices satisfying AB = I(sub)n then A^-1 = B and B^-1 = A

    Well, if AB = I then rk(B) >= rk(AB) = rk(I) = n, but clearly rk(B) <= n for any nxn square matric ==> rk(B) = n and B is invertible. The same is true for A and thus from AB = I we get A, B are inverses of each other.

    Tonio

    all i know so far is that rank is the number of pivot columns of a matrix
    idk why our prof is introducing if then statements now, it's the last chapter in our textbook!
    anyway, help would be much appreciated!
    .................................................. ...
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  3. #3
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    given subspaces  V, W \ \in R^2

    Thinking the intersection in R^2, it could be only two things: a line (when both subespaces were the same line) or a point (0,0), when both were different.
    (Remember that the (0,0) must be in any subspace).

    in both cases you get a subspace.

    The union could be: the same line (when both were the same line) or two different lines.

    In the first case  \ V\ \cup \ W is, of course a subspace.

    In this case  \ V \ \cup \ W is not a subspace since you can add two element on it and get an element from the outside.


    viko
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  4. #4
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    Thanks guys, very helpful!

    I guess I didn't get lucky on the bonus... oh well!
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