I\'m having trouble with the following problem. Any help you can give me would b

Question

I'm having trouble with the following problem. Any help you can give me would be great.

Consider a non-zero vector v in R^3. Arguing geometrically, describe the image and kernel of the linear transformation T from R^3 to R^3 given by: T(X) = v x X            (A.k.a. T(X) is the cross product of the vectors v and X)
Thanks in advance!
Consider a non-zero vector v in R^3. Arguing geometrically, describe the image and kernel of the linear transformation T from R^3 to R^3 given by: T(X) = v x X            (A.k.a. T(X) is the cross product of the vectors v and X)
Thanks in advance!

Explanation / Answer

As the problem suggests, think geometrically. The cross product of v with any vector x gives a vector perpendicular to v and x; in particular, it is perpendicular to v. Hence if you start with a vector v, look at the orthogonal complement of v. This is a plane orthogonal to v, and the cross product must lie in this plane. Conversely, for any vector w in this plane, it is easy to see that for some vector x, w is the cross product of v and x. Hence the image of this map T is the orthogonal plane to v. v x X = 0 if and only if X is parallel to v, i.e., X = av for some constant a. Hence the kernel is the vector space generated by v.

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