# LINEAR ALGEBRA - Describe the kernel of a linear transformation GEOMETRICALLY

• VinnyCee
In summary: The geometric relationship between \vec{x} and \vec{u},\vec{v} is that the vectors are perpendicular if and only if det[\overrightarrow{x}\,\,\overrightarrow{v}\,\,\overrightarrow{w}\right] = 0.
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"?

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.

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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.

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).

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VinnyCee said:
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.

The formula $L_y : x \mapsto x \wedge y$ says that Ly 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 Ly is the set of vectors {x | Ly(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 Cy 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?

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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?

## 1. What is the kernel of a linear transformation?

The kernel of a linear transformation is the set of all vectors that are mapped to the zero vector by the transformation. In other words, it is the set of all inputs that result in an output of zero.

## 2. How is the kernel of a linear transformation related to the null space?

The kernel of a linear transformation and the null space are essentially the same concept. They both refer to the set of vectors that are mapped to zero by a linear transformation. The only difference is that the term "null space" is typically used when referring to a matrix, while "kernel" is used when referring to a transformation.

## 3. Can you give an example of the kernel of a linear transformation?

Yes, for a linear transformation T: R^2 -> R^2, where T(x,y) = (x+y, x-y), the kernel would be the line y=x, since any vector on this line would be mapped to zero by the transformation.

## 4. How can the kernel of a linear transformation be described geometrically?

The kernel of a linear transformation can be described geometrically as the set of all vectors that lie on a specific subspace, such as a line, plane, or hyperplane, depending on the dimension of the vector space.

## 5. What is the significance of the kernel in linear algebra?

The kernel of a linear transformation is important because it helps us understand the behavior of the transformation and its relationship to the vector space. It also allows us to find solutions to systems of linear equations and determine if a transformation is invertible.

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