Image
Kernal
Linear Independence
Basis
Dimension
Rank-Nullity Theorem
Coordinates How to Examples Exercise
How to:
The dimension of a subspace V of Rn is the number of vectors in a basis needed to span V. It's that simple, once we find a basis for V, all we have to do is count.
Examples:
In the last section we were working with the matrix R = [3 -1 4; 4 -2 6; 5 -3 7]. After some routine calculations, we found the basis for R's image to be the vectors [3; 4; 5] and [4; 6; 7].
Here our image is a subspace of R3 since we are dealing with 3x1 column vectors. Counting the number of vectors in its basis, we see the image's dimension is 2, or a plane in R3.
Exercise: