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

 P: 492 1. The problem statement, all variables and given/known data 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? 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"?
 P: 492 LINEAR ALGEBRA - Describe the kernel of a linear transformation GEOMETRICALLY 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).
 Sci Advisor HW Helper P: 2,586 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?
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 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?