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Thread: Please verify.

  1. #1
    Aug 2008

    Wink Please verify.

    If \vec{a}\times 2\vec{j}=0, what can you say about \vec{a}?
    \vec{a} and \vec{j} are parallel; paralelogram they span has area 0

    There's gotta be more?? Please tell me.

    If \vec{b}\times\vec{j}=2\vec{i} and \langle \vec{b}, \vec{j}\rangle=-1, what's \vec{b}?
    Are \vec{i} and \vec{j} meant as orthonormal vectors (1,0,0) and (0,1,0)?

    If so, is this right: \vec{b}\times\vec{j}=(-b_3)\vec{i}+b_1\vec{k}=2\vec{i} \Rightarrow b_3=-2, b_1=0, b_2\in\mathbb{R}\Rightarrow \vec{b}=(0,p,-2);p\in\mathbb{R}??? Geometrically sounds right, in right-handed system \vec{-k}\times\vec{j}=\vec{i}, but I want to be sure.

    If \vec{c}=(c_1,c_2,c_3) and \vec{d}=(-1,2,0) what is \vec{c}\times \vec{d}?

    \vec{a}=(1,1,1,1), \vec{b}=(1,-1,1,-1), \vec{c}=(1,i,-1,-i)Are \vec{c},\vec{c},\vec{c} in-dependent? Do they make a basis in \mathbb{C}^4
    Please, check my scalar products:
    \langle c,c\rangle=1-1+1-1=0

    \langle a,b\rangle=1-1+1-1=0
    \langle a,c\rangle=1+i-1-i=0
    \langle b,c\rangle=1-i-1+i=0
    So, they are in-dependent.

    But, they don't make a basis because "there's not enough of them" () for 4D vector space; they don't span the whole space? OK, what's more rigorous answer?

    Let X=Y=\mathbb{R}^2 and V be rotation of 90 around origin. Define kernel, image and rank of map V.
    In kernel is only \vec{0}, right?
    Image is the whole \mathbb{R}^2???
    Rank of V must be 2, otherwise it couldn't rotate (how could I better formalise my answer; if the 2 is correct of course)?

    What should X=Y=\mathbb{R}^2 mean (what are X and Y, obviously not points)?!

    Let Y be linear sub-space inside \mathbb{K}^n. Define orthogonal projection \vec{y} of \vec{z}\in\mathbb{K}^n to Y.
    (Connects with previous quote.) Let also \big{\vec{f_1},...,\vec{f_k}\big} be ortho-normal basis for Y. Express \vec{y} projection (from previous) with this basis.
    Last edited by courteous; Sep 14th 2009 at 03:01 PM. Reason: added sub-space questions
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