Monthly Archives: November 2008

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
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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.

Note on quadratic forms

For those of you who downloaded the note on quadratic forms from the 2201 course website, you should be aware that there is a revision now available. Some misprints were corrected as well as a significant error where the inverse of the transition matrix should have been used.

Change of basis for linear maps and bilinear forms

dear professor,

when I read the note, I find these problems, hope you do not mind

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1. in page 40, at the end of 4.1.100 proposition, you wrote “and for any linear map T:V \rightarrow V we have \left[T\right]_C= M^{-1}\left[T\right]_B M,” can you explain why? because in 4.1.101, the note said \left[f\right]_C = M^{t}\left[f\right]_B M, one is M^{-1} and one is M^{t} which makes me a little confused.

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2. in page 40, I think all the appearance of “theorem 5.1” should be changed to “theorem 4.1”. am I right?
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thanks very much
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Reply:

1. The intention of that paragraph was exactly to warn you that the way the matrix of a bilinear form changes is different from the way the matrix of a linear map changes when we go from one basis to another.

2. Yes, it should have been theorem 4.1. Thanks very much for the correction.

Vector question

Dear Professor.

I want to ask you a maths question. Hope you don’t mind.

What is the difference between

|C| = \frac {A \cdot B} {|B| },

and

C = \frac {A \cdot B} {|B|^2} B ?

Where C is the projection of A on B.

I understand that the first expression is the magnitude of C, which is the length of the projection, am i right?

However, I’m uncertain about the second expression and to what significance it holds. Please enlighten me regarding this.

Thank you very much! Your help is much appreciated.

Take care!

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Reply:

You are right about the length of the projection. Perhaps the best way to see the meaning of the second expression is to write it as:

C=(A \cdot (\frac{B}{|B|}))\frac{B}{|B|}

Here is the point: Given any vector W, I believe you know how to visualize its projection along any directed line L. For simplicity, I will assume that W forms an acute angle with the direction of the line. The projection is then the shadow of W you see when you shine a light on L along a perpendicular angle. It is an elementary exercise with a triangle to see that the length of this projection is exactly

W\cdot e_L,

where e_L is a *unit vector*, that is, a vector of length 1, in the direction of L. You should think of a unit vector as encoding ‘pure direction.’ So to get a vector of magnitude r in the direction of the unit vector E, you can just dilate E by a factor of r to get

X=rE.

Conversely, to change any non-zero vector V into the unit vector in the same direction, you divide by its magnitude to get

\frac{V}{|V|}.

Note now the key fact that the projection of W itself, not just its magnitude, is given by the formula

(W\cdot e_L)e_L

since the projection is clearly in the direction of e_L and this expression gives such a vector with just the right magnitude.

To return to your question, given any two vectors W and V, you can project W in the direction of V. Now, the unit vector in the direction of V is exactly

\frac{V}{|V|}.

So the projection of W in the direction of V is nothing but

(W\cdot (\frac{V}{|V|}))\frac{V}{|V|}.

By the way, your question is a very good one. It’s very important to understand the geometric meaning of basic expressions like this. Otherwise, everything degenerates into a bunch of formulas.

Integral bases

Because my lecture just before the break was rather vague about the algorithm for computing integral bases, I’ve written up a note that explains it in greater detail. Here is a link.