# Finding possible values for dimension of image and kernel

Say I have a 3x5 matrix that represents a linear transformation from $$R^5$$ to $$R^3$$. I want to find possible values of the dimension of the kernel, and the possible values for dimension of the image. I know that the maximum possible rank is 5, and I know the rank-nullity theorem, but I'm not sure how exactly to apply it here. Thanks!

Actually, you're wrong when you say that the maximum possible rank is $$5$$. Since the rank is the dimension of the image and since the image is a subspace of $$\mathbb{R}^3$$, the maximum possible rank is $$3$$. And, by the rank-nullity theorem, and since the dimension of the domain is $$5$$, the dimension of the kernel is $$5$$ minus the rank. So, in principle, there are $$4$$ possibilities:

• rank $$3$$ and dimension of the kernel $$2$$;
• rank $$2$$ and dimension of the kernel $$3$$;
• rank $$1$$ and dimension of the kernel $$4$$;
• rank $$0$$ and dimension of the kernel $$5$$.

Now, find an example for each case (Hint: for the last case, there is just one possible example.)