Good day to all. I have just read the section in my textbook on rank, kernel and

Question

Good day to all.
I have just read the section in my textbook on rank, kernel andlinear dependence/independence. I thought I understood the chapterbut as I was doing the end of section exercises, I realized thatthere is something I have missed. The question is as follows:

Question:

Find a matrix whose kernel is spanned by the vectors u = (1,3,2)and v = (-2,0,4)


Does this mean that some linear combination of u and vsay w = au + bv (a, b are real numbers) will produce Aw = 0 ??

Any helpwould be greatly appreciated!

Explanation / Answer

indeed. suppose M is a 3x3 matrix which is reduced to the echlon formand one of the rows have become completely zero. then we write homogeneous equations from the reduced matrixand solve them to get the column vectors which are now given. these column vectors will be the basis of the null space of Mor the kernel of M. while the vector is a triplet and there are two vectors in thebasis of the kernel , its dimension is 2 and so, the dimension ofthe range space is 1.

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