Abstract Nonsense

Crushing one theorem at a time

Invariant Subspaces


Point of Post: In this post we’ll discuss the equivalent of section 39 in Halmos, but in more detail.

 

Motivation

Often in mathematics it’s fruitful to take a look at a class of commonly occurring, easily dealt with set of examples to enrich the overall tapestry of one’s knowledge. Linear transformations are no-different. We’d like to study the idea now of invariant subspaces. The idea being that if T is a linear homomorphism with a non-trivial invariant subspace then T is ‘nicer’ in some sense. We make this more rigorous below.

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December 20, 2010 Posted by | Algebra, Halmos, Linear Algebra, Uncategorized | , | 3 Comments

Halmos Section 36: Inverses (Pt. III)


Point of post: This is a continuation of this post.

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December 4, 2010 Posted by | Fun Problems, Halmos, Linear Algebra | , , , , , | Leave a comment

Halmos Section 36: Inverses (Pt. II)


Point of post: This is a continuation of this post.

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December 3, 2010 Posted by | Fun Problems, Halmos, Linear Algebra, Uncategorized | , , , , , | 1 Comment

Characterization of Linear Homomorphisms In Terms of Bases


Point of post: In this post we give a characterization of linear homomorphisms which has to do entirely with how the homomorphism acts on a basis.

Motivation

Often it is laborious to check that that a linear transformation is injective or surjective. It turns out that there is a nice characterization of injectivity and surjectivity for finite dimensional spaces which depends entirely on how the the transformation acts on a basis of the domain space.

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December 2, 2010 Posted by | Algebra, Halmos, Linear Algebra | , , , | 3 Comments

Invertible Linear Transformations (Pt. I)


Point of post: In this post we talk about the equivalent of Halmos’s section 36, but in a more general setting.

Motivation

Recall that in our last post we discussed how to turn \text{End}\left(\mathscr{V}\right) into an associative unital algebra by defining a ‘multiplication map’

\mu:\text{End}\left(\mathscr{V}\right)\times\text{End}\left(\mathscr{V}\right)\to\text{End}\left(\mathscr{V}\right):\left(T,T'\right)\mapsto T\circ T'

We saw though that this multiplication has some downsides (it had zero divisors and wasn’t, in general, commutative).  There is a nice property of this algebra which isn’t enjoyed by all algebras. In particular, it is not ‘uncommon’ for elements of \text{End}\left(\mathscr{V}\right) to have multiplicative inverses, in the usual sense.  Now, at first glance this doesn’t seem that ‘great’ of a property; I mean, not all of  the elements of \text{End}\left(\mathscr{V}\right) have multiplicative inverses just some do. One may expect that, in general, this is a common occurrence among algebras, even more common among associative unital algebras. In fact, this isn’t the case. To see how badly the existence of multiplicative inverses can get screwed up consider for example the polynomial ring \mathbb{R}[x]. It’s fairly plain to see that \mathbb{R}[x] is, in fact, an associative unital algebra over \mathbb{R} with the usual polynomial addition and multiplication. That said, a little thought shows that p(x)\in\mathbb{R}[x] has a multiplicative inverse only if p(x)\in\text{span}\{1\}.

Thus, this post will explore this ‘nice’ quality of the multiplication map on \text{End}\left(\mathscr{V}\right)

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November 30, 2010 Posted by | Algebra, Halmos, Linear Algebra | , , , , , , , | 7 Comments

Halmos Sections 32 and 33: Linear Transformations and Transformations as Vectors (Pt. II)


Point of post: This is a continuation of this post in an effort to answer the questions at the end of sections 32 and 33 in Halmos’s book.

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November 22, 2010 Posted by | Fun Problems, Halmos, Linear Algebra, Uncategorized | , , , , | Leave a comment

Halmos Sections 32 and 33: Linear Transformations and Transformations as Vectors (Pt. I)


Point of post: In this post we complete the problems that appear at the end of Halmos, sections 32 and 33.

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November 22, 2010 Posted by | Fun Problems, Halmos, Linear Algebra | , , , | 3 Comments

Linear Transformations


Point of post: In this post I discuss the material discussed in sections 31 and 32 of Halmos.

Motivation

We finally begin studying arguably the most important aspect of linear algebra, linear transformations. These are the structure preserving maps for vector spaces. They correspond naturally to coordinate changes, linear shifts, and contraction/dilations. Moreover, theoretically they are interesting because, as it turns out, many maps which aren’t usually thought of as linear transformations are. For example, the differential operator on the space C^{\infty}[\mathbb{R},\mathbb{R}].

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November 19, 2010 Posted by | Algebra, Halmos, Linear Algebra, Uncategorized | , , , | 8 Comments