- #1

*FaerieLight*

- 43

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I'm given a matrix,

[2,4,1,2,6; 1,2,1,0,1; ,-1,-2,-2,3,6; 1,2,-1,5,12], which is the matrix representation of a linear operator from R

^{5}to R

^{4}.

The question asks me to find a basis of the image and the kernel of the map.

Row-reducing the matrix gives

[1,2,0,0,-1; 0,0,1,0,2; 0,0,0,1,3; 0,0,0,0,0,] as the reduced row echelon form. I'm told that the columns 1, 3 and 4 are linearly independent, and hence the corresponding columns of the original matrix form a basis of the image of A. This is what I don't understand. How do I know that these columns are linearly independent? I suppose it has something to do with the fact that these columns are the only ones with a single 1 entry, while all the rest of the entries are 0, above and below. But why should this feature necessarily mean that the columns are linearly independent?

After determining the linearly independent columns, why should the corresponding columns of the original matrix correspond to a basis of the image of A?

Also, how would I find a basis for the kernel from the row-reduced echelon form?

Thanks very much.