91影视

The kernel of linear transformation is defined as follows:

The kernel of a linear transformation $T\left(\stackrel{鈫�}{x}\right)=A\stackrel{鈫�}{x}$ from ${\mathrm{鈩�}}^m$to ${\mathrm{鈩�}}^n$consists of all zeros of the transformation, i.e., the solutions of the equations $T\left(\stackrel{鈫�}{x}\right)=A\stackrel{鈫�}{x}=0$.

It is denoted by ker$\left(T\right)$or ker$\left(A\right)$

Thedefinition A.9 cross productin ${\mathrm{鈩�}}^3$is given as follows:

The cross product $\stackrel{鈫�}{蠀}脳\stackrel{鈫�}{蝇}$of two vectors role="math" localid="1659527572946" $\stackrel{鈫�}{蠀}$and $\stackrel{鈫�}{蝇}$in ${\mathrm{鈩�}}^3$is the vector in ${\mathrm{鈩�}}^3$with three properties as follows:

1. $\stackrel{鈫�}{蠀}脳\stackrel{鈫�}{蝇}$is orthogonal to both $\stackrel{鈫�}{蠀}$and$\stackrel{鈫�}{蝇}$.
2. $||\stackrel{鈫�}{蠀}脳\stackrel{鈫�}{蝇}||=||\stackrel{鈫�}{蠀}||\sin 胃||\stackrel{鈫�}{蝇}||$, where $胃$is angle between $\stackrel{鈫�}{蠀}$and $\stackrel{鈫�}{蝇}$with $0猢�胃猢�蟺$.
3. The direction of $\stackrel{鈫�}{蠀}脳\stackrel{鈫�}{蝇}$follows the right-hand rule.

We have given the linear transformation $T:{\mathrm{鈩�}}^3鈫�{\mathrm{鈩�}}^3$defined by $T\left(\stackrel{鈫�}{x}\right)=\stackrel{鈫�}{蠀}脳\stackrel{鈫�}{x}$,

With the reference of definition in step 1, we have,

$T\left(\stackrel{鈫�}{x}\right)=\stackrel{鈫�}{蠀}脳\stackrel{鈫�}{x}=0^U\stackrel{鈫�}{x}=位\stackrel{鈫�}{蠀},鈥�鈥�鈥�位^IR$is any scalar, which means that the cross product is zero.

This implies that kernel is a line in space spanned by vector$\stackrel{鈫�}{蠀}$