LINEAR ALGEBRA - Describe the kernel of a linear transformation GEOMETRICALLY

  1. 1. The problem statement, all variables and given/known data

    For two nonparallel vectors [itex]\overrightarrow{v}[/itex] and [itex]\overrightarrow{w}[/itex] in [itex]\mathbb{R}^3[/itex], consider the linear transformation


    from [itex]\mathbb{R}^3[/itex] to [itex]\mathbb{R}[/itex]. Describe the kernel of T geometrically. What is the image of T?

    2. Relevant equations

    I have no idea. Maybe the equations on how to find a kernel and image?

    3. The attempt at a solution

    I don't know where to even start this exercise! How does one "describe geometrically"?
  2. jcsd
  3. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    Given v and w, when does it vanish? No equations, nothing like that, just a simple statement of what it means when det vanishes. If you just use words, you'll be describing it geometrically. For instance, fix a y, and take the linear map

    L_y : x--> x /\y

    which takes x and sends it to the vector product of x and y, then the kernel is the set of x that are parallel to y (or the line spanned by y). That is a geometrical description of the kernel.

    The point is that you could let x=(x_1,x_2,x_3) and v=(v_1,v_2,v_3) etc and write down an equation f(x_1,x_2,x_3)=0 with coefficients the v_i, w_i which parametrizes the kernel, but it would be incredibly unhelpful when there is a far simpler description.
    Last edited: Dec 3, 2006
  4. radou

    radou 3,108
    Homework Helper

    What is the definition of a kernel? How does that apply in your case? What is the geometrical representation of that?

    Edit: too late, again.
  5. Thank you for trying to explain this concept to me, however, I still do not understand!

    Can you explain the formula [tex]L_y: x\,->\,x\,\bigwedge\,y[/tex]?

    Is that also expressed as the "dot product"? The left of the equation reads "Linear transformation of y", right? [tex]L_y:\,x\,->\,\overrightarrow{x}\,\cdot\,\overrightarrow{y}[/tex]

    Maybe if you just explain it in very precise terms that a "lay-person" would understand? I always have trouble with these dang kernels!

    I know that a kernel is the functions or vectors that cause the transformation to be equal to zero (at least, I hope it is).
    Last edited: Dec 3, 2006
  6. radou

    radou 3,108
    Homework Helper

    More precisely, the kernelis the set of vectors that 'cause' the transformation to be equal to zero.
  7. AKG

    AKG 2,576
    Science Advisor
    Homework Helper

    The formula [itex]L_y : x \mapsto x \wedge y[/itex] says that Ly is a function that maps x to [itex]x \wedge y[/itex]. The wedge product is a generalization of the cross product, not the dot product. The kernel of Ly is the set of vectors {x | Ly(x) = 0}, which is exactly the same thing as [itex]\{ x\, |\, x \wedge y = 0\}[/itex]. Like I said, the wedge product is just a generalization of the cross product, so it's probably easier for you to consider instead the function Cy defined by [itex]C_y : x \mapsto x \times y[/itex]. Then:

    [tex]\mbox{Ker}(C_y) = \{ x\, |\, C_y(x) = 0\} = \{ x\, |\, x \times y = 0\}[/tex]

    This set is obviously just the set of vectors perpendicular to y, because [itex]x \times y = 0[/itex] iff x and y are perpendicular. You know that, right?
    Last edited: Dec 4, 2006
  8. HallsofIvy

    HallsofIvy 41,264
    Staff Emeritus
    Science Advisor

    This might help: For three vectors, [itex]\vec{x},\vec{u},\vec{v}[/itex],
    also called the "triple" product, is [itex]\vec{x}\cdot\left(\vec{u} X \vec{v}\right)[/itex].
    Of course, the dot product of two vectors is 0 if and only if they are perpendicular, and the cross product of two vectors is perpendicular to both of them. What does that tell you about the geometric relationship between [itex]\vec{x}[/itex] and [itex]\vec{u},\vec{v}[/itex] if this is equal to 0?
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