## Interesting Linear Algebra and Matrix Analysis Problems

**Point of post:** In this post I will just do some neat problems from basic linear algebra and matrix analysis. Feel free to comment/critique any.

**NOTE:** It would be very appreciated if anyone could give me some more linear algebra problems to do.

1.

**Problem:** Let be a -dimensional -space and denote it’s scalar multiplication by and a subfield of . Then:

**a) ** is a vector space over with usual addition and multiplication.

**b) **If then is a vector space over with multiplication and

**Proof:**

**a) **This is immediate from the field axioms.

**b) **The fact that over forms a vector space is clear. Now, to prove the claim about the dimensionality let be a basis for over and a basis for over . We claim that is a basis for over . To see this, first suppose that were such that

Then, by the linear independence of we see that . But, since are are linearly independent over we see that each of the sums above implies that from where the linear independence of follows.

To see that we may merely notice that since for any there exists such that . But, since we know that where . So, then upon insertion and expansion one arrives at

where and

2.

**Problem:** Let . Show that

**Proof:** This is equivalent to asking why is not equal to a polynomial. But, this is clear since if is any polynomial then but (where is the differential operator).

3.

**Problem:** Let be a field. Show then that if one considers as a vector space over itself that there does not exist proper subspaces and such that .

**Proof: **The obvious way is to note that if that , and clearly this implies that, up to relabeling, and . Thus, since is a -dimensional subspace of (which is itself -dimensional) we may conclude that . Similarly, since we may conclude that .

4.

**Problem: **Show that if is endowed with the usual topology and is a subspace of (with the usual operations) then is closed.

5.

**Problem:** Let be additive () and continuous.. Prove that is a linear functional.

**Proof:** First notice that it follows by induction that

for all . Furthermore, if we see that

Thus,

So, by combining these two we see that if that

Thus, if we see that

Thus, if

we see that . But, this means that and are continuous maps which agree on a dense subset of the domain, which means that they must agree on all of the domain. Namely, . But, is evidently a linear functional and so the conclusion follows.

6.

**Problem:** Let be both normal and nilpotent. Then

**Proof:** By the spectral theorem we have that

for some unitary matrix . Thus,

Thus,

and so

But, using normality and nilpotence we see that

so, upon comparision we find that

and so . Thus, appealing to we find that

7.

**Problem:** Let be such that for some . Show that for some .

**Proof: **Note that if (where is the spectrum of a matrix) with associated eigenvector , then

and since we may conclude that . Thus, all the eigenvalues of are . But, then we know that

where

is the Jordan block associated with the particular eigenvalue. But, since we evidently have that . But, evidently if we have that

and so

which is what was to be proven.

8.

**Problem:** What are the possible Jordan canonical forms for given that ?

**Proof: **We know that

for some Jordan matrix . Thus,

But, notice that if

that

And thus, for it’s evident that . But, if $n_{\ell}>0$ it easily follows that (just consider induction on and partition the matrix into smaller ones). Thus, it follows that so that . Thus, we may conclude that

where are eigenvalues of , and thus -th roots of unity.

**References:**

1. Halmos, Paul R. *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print.

2. Brown, William C. *A Second Course in Linear Algebra*. New York: Wiley, 1988. Print.

3. Horn, Roger A., and Charles R. Johnson. *Matrix Analysis*. Cambridge [u.a.: Cambridge Univ., 2009. Print.

4. Roman, Steven. *Advanced Linear Algebra*. New York: Springer, 2005. Print.

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