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Well, I just finished my first Oxford tutorial - on Linear Algebra. This being the case, I thought perhaps I would ask myself “What is Linear Algebra?” and then try to explain it a bit, so that someone might be able appreciate what is I’ve been doing with my life for the past two months.

Better yet, I won’t explain it but instead use a bunch of fancy symbols in some examples below. Primarily, I discovered that you can use all of these mathematical symbols in HTML, so I needed an excuse to try writing some stuff out. This was the perfect excuse.

Definition of the Kernel

Let V, W be vector spaces over some field, F, and let v ∈ V. If T:V→W is a linear transformation, define Ker(T) = { v ∈ V s.t. T(v) = 0_W }.

Example of An Inner Product

Let V be the vector space consisting of all real n×n matrices, that is, let V = M_n×n(ℜ). ∀A,B ∈ V, define <A,B> := Trace(AB^tr). Show that {V, <,>_V } is an inner product space on V.

Response: ∀ α,β ∈ ℜ, ∀ A,B,C ∈ V,

1. <αA+βB, C> = Trace((αA+βB)C^tr)
       = Trace(αAC^tr+βBC^tr)
       = Trace(αAC^tr) + Trace(βBC^tr)
       = αTrace(AC^tr) + βTrace(BC^tr)
       = α<A,C> + β<B,C>.
2. <A,B> = Trace(AB^tr)
       = Trace((AB^tr)^tr)
       = Trace(BA^tr)
       = <B,A>.
   So <A,B> is symmetric, and hence is bilinear.
3. If A ≠ 0, ∃ a_ij∈A s.t. a_ij ≠ 0. Also, AA^tr = Σa_ija_ji. For b_ii ∈ AA^tr, b_ii ≥ Σa_ii² > 0. So <A,A> is positive-definite. This completes the proof.

Matrix Representation of An Self-Adjoint Transformation

Let { e₁, e₂ … e_n } be an orthonormal basis of a vector space V over ℜ, and T:V→V be a self-adjoint linear transformation. Now let A be the matrix of T w.r.t. { e₁, e₂ … e_n }, and write A = (a_ij).

We compute T(e_i) = Σa_jie_j on j, so that when <T(e_i), e_k> = <e_i, T(e_k)>, we see that:

1. <T(e_i), e_k> = <Σa_jie_j, e_k> = Σa_ji<e_j,e_k> = a_ki.
2. <e_i, T(e_k)> = <e_i,Σa_jke_j> = Σa_jk<e_i,e_j> = a_ik.

T is self-adjoint ⇔ a_ik = a_ki ⇔ A = A^tr, so A is symmetric.

Observe that if V is a vector space over the complex field; then A = conj(A)^tr.

Tags: algebra, basis, bilinear, complex, html, inner product, IPS, kernel, linear algebra, linear transformation, matrix, orthonormal, Oxford, positive-definite, self-adjoint, symbol, symmetric, trace, transformation, tutorial, vector space

This entry was posted on Friday, November 30th, 2007 at 8:40 pm and is filed under Oxford.