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A Question on Notation about Linear Algebra

  1. Jan 2, 2012 #1
    Could I write
    v = 4i + 3j -2k as
    v =
    [ 4 |
    | 3 | * I_3
    | -2 ]

    were I_3 is the 3x3 identity matrix and the other thing it's multiplied by is a 3x1 matrix of values 4,3,-2

    I also had a question about the notation
    f(x) = x^2 + 3
    can be written as
    f: x |-> x^2 + 3

    can I write vectors in this matter as well if instead
    v = 4x^2 i + 3x j -2x k
    as
    v: x |-> 4x^2 i + 3x j - 2x k
    or something like
    v: x |->
    [ 4x |
    | 3 | * x * I_3
    | -2 ]

    I'm sort of confused by
    f: x |-> and just want to make sure I'm applying it correctly and am wondering if I can use the identity matrix instead of the unit vectors?
     
  2. jcsd
  3. Jan 2, 2012 #2

    Mark44

    Staff: Mentor

    Sure, but I think it adds needless complexity.
    This looks OK. Your notation is saying that v maps a number x to a vector-valued function.
    I don't see anything wrong with this, but it seems like overkill.
     
  4. Jan 2, 2012 #3
    Can I write
    v = 4i + 3j -2k
    as v =
    [ 4 |
    | 3 |
    | -2 ]
    or this way
    v = [ 4 3 -2]?

    I believe this writing it like this would take less time. All you need is three numbers and two brackets [ ]. Writing it like this 4i + 3j -2k requires you write three numbers, three unit vectors, and any plus signs that might be needed. I'm just not sure that
    v = [ 4 3 -2] = 4i + 3j -2k is true or maybe if it was a 3x1 matrix instead of a 1x3 this be true? If either of these are I rather would because it's less writing.
     
  5. Jan 3, 2012 #4

    Mark44

    Staff: Mentor

    I agree with you about the i, j, k notation taking more effort to write, and usually write vectors in this form: <4, 3, -2>. If they really need to be column vectors, you can add a T superscript (for transpose), like this: <4, 3, -2>T.
     
  6. Jan 3, 2012 #5
    I hate the i, j, k mainly because my handwriting is terrible and I find i and k to be incredibly hard to write without making everything look messy

    nono, it's symbol, domain, codomain, correspondance
    [itex]f: \Re \rightarrow \Re ; x \rightarrow x^2+3[/itex]
    the script R is both the doman and the codomain (R+, the positive reals is technically the codomain(technically the complex numbers is the domain and the codomain))
    the x -> x^2 + 3 is the rule of correspondance

    if you really wanted to you could but there isn't really much point since you'd end up writing 6 extra zeros

    Also you can write [itex]\begin{pmatrix} x \\ y \\ z \end{pmatrix}[/itex] as [itex]\begin{pmatrix} x & y & z\end{pmatrix}^T[/itex] or [itex]\begin{pmatrix} x \\ y \\ z \end{pmatrix}^T[/itex] as [itex]\begin{pmatrix} x & y & z\end{pmatrix}[/itex]
     
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