linear algebra questions

Dear Professor,

I’m a second year Algebra student, and I was hoping you could answer a couple of questions.

1.

a. Notation. In the online notes, page 43, you refer to the set GL_n (k). I don’t recall seeing this notation before – what set does this refer to?

b. On page 45 you twice refer to q(x,y)_t seeming to mean a transposed quadratic form. What does this mean? How can one transpose what is essentially a function?

2.

From my notes from Thursday’s lecture, in an example we consider a quadratic form q, C_1 the standard basis for R^3, C_2 a different basis, and M the change of basis matrix from C_1 to C_2, so that \left[q\right]_{C_1} = B, B being a 3×3 matrix which we had earlier diagonalised via double operations to obtain the diagonal matrix D.
We called the matrix that effected those double operations P, so that

P^TBP = D.

Then, changing basis from C_1 to C_2, we wrote:

\left[q\right]_{C_2} = M^t\left[q\right]_{C_1}M

Then the following line is what I don’t understand: “Because C_1 is the standard basis for k^3, the columns of P are exactly elements of the new basis C_2” Why is this the case?

Apologies for the slightly involved question, I am more than happy to come and explain the problem in person – I intended to come to the office hour today, but unfortunately forgot my notes… (!)

Thanks
——————————————————————————————

Reply:

1. a. GL_n(k) refers to the set of n\times n invertible matrices with entries in k. It actually forms a group under multiplication. Taking k to be R or C give the most basic examples of Lie groups.

b. This is slightly odd notation, but I’ll let it stand. The point is that we usually view vectors in k^n as column vectors. So I wrote

q(x,y)^t

to mean “the function q of the column vector (x,y)^t.”

2. There are two facts: If the change of basis matrix from C_1 to C_2 is M, then

\left[q\right]_{C_2} = M^t\left[q\right]_{C_1}M.

Conversely, if C_1 is a basis, then for any invertible matrix M,

M^t\left[q\right]_{C_1}M

is the matrix of q with respect to the basis C_2 with change of basis matrix M from C_1. So in our case, D is the matrix of the quadratic form with respect to the basis C_2 with the property that the change of basis matrix from C_1 to C_2 is P. However, how does one calculate the change of basis matrix M from the standard basis

C_1=\{e_1, e_2, e_3\}

to a basis C_2=\{b_1, b_2, b_3\}? The first column of M is given by the coefficients in the expression of b_1 as a linear combination of the e_i. But these coefficients are nothing but the entries of b_1. Thus, b_1 is exactly the first column of M. Similarly for the other columns. So in our case, when we pose the question of “what is the basis with respect to which q acquires the diagonal form D?”, the answer is given by the columns of the change of basis matrix P.

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