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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!

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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.)

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