LINEAR ALGEBRA - Describe the kernel of a linear transformation GEOMETRICALLY

[VinnyCee]

Homework Statement

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

$$T\left(\overrightarrow{x}\right)\,=\,det\left[\overrightarrow{x}\,\,\overrightarrow{v}\,\,\overrightarrow{w}\right]$$

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

Homework Equations

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

The Attempt at a Solution

I don't know where to even start this exercise! How does one "describe geometrically"?

 

[matt grime]

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.

 

[radou]

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.

 

[VinnyCee]

Thank you for trying to explain this concept to me, however, I still do not understand!

Can you explain the formula $$L_y: x\,->\,x\,\bigwedge\,y$$?

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

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).

 

[radou]

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).

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

 

[AKG]

The formula $L_y : x \mapsto x \wedge y$ says that L_y is a function that maps x to $x \wedge y$. The wedge product is a generalization of the cross product, not the dot product. The kernel of L_y is the set of vectors {x | L_y(x) = 0}, which is exactly the same thing as $\{ x\, |\, x \wedge y = 0\}$. 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 C_y defined by $C_y : x \mapsto x \times y$. Then:

$$\mbox{Ker}(C_y) = \{ x\, |\, C_y(x) = 0\} = \{ x\, |\, x \times y = 0\}$$

This set is obviously just the set of vectors perpendicular to y, because $x \times y = 0$ iff x and y are perpendicular. You know that, right?

 

[HallsofIvy]

This might help: For three vectors, $\vec{x},\vec{u},\vec{v}$,
$$det\left[\overrightarrow{x}\,\,\overrightarrow{v}\,\,\overrightarrow{w}\right]$$
also called the "triple" product, is $\vec{x}\cdot\left(\vec{u} X \vec{v}\right)$.
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 $\vec{x}$ and $\vec{u},\vec{v}$ if this is equal to 0?