SUBSPACES OF Rn AND THEIR DIMENSIONS

Image Kernal Linear Independence Basis Dimension Rank-Nullity Theorem Coordinates

Basis How to Examples Exercise How to: A basis is a set of vectors that span some subspace of Rn and are all linearly independent. In the Image section, we discussed how the vectors of a matrix span some space in codomain called the image. But the vectors of this matrix do not necessarily form a BASIS for the image. They do meet one of the requirements, spanning the image, but we first need to check if they are linearly independent before we can term them a basis.      Examples: Let's find the image and then a basis for the image of the matrix, R = [3 -1 4; 4 -2 6; 5 -3 7]. Finding the image is simple, it is the span of the column vectors of R, namely span([3; 4; 5], [-1; -2; -3], [4; 6; 7]). But is this a basis as well? No, it is not. When we look at rank(R), we see that is equals 2, even though R has 3 columns. So, one of them is a combination of the others. So, we now find rref(R). In RREF, rref(R) = [1 -1 0; 0 0 1; 0 0 0]. So, the middle column ([-1; 0; 0]) is a combination of the other two, and the basis for our image are the vectors [3; 4; 5] and [4; 6; 7]. Exercise: