« Mid-term Solutions and Analysis
Questions concerning Magic Squares and Matrices »
h1

Hints to Selected Problems in HW 3

二月 5, 2009

Many of you were stuck in problem 1, and I think part of the reason is that you have trouble understanding the symbols.

For example, here is the definition of ker(A) = \{ v \in R^n \, | \, Av = 0 \}. This literally translate into:

ker(A) is the set of vectors in R^n such that Av = 0. Basically, it tells you a criterion to test whether a vector is in ker(A), and you can also interpret it as

v \in ker(A) if Av = 0.Also, in writing mathematical proofs, we always have three components:

1) Assumptions /Hypothesis

2) Reasoning/Proof.

3) Conclusion.

Therefore, you should be constantly asking yourself the following questions when you are writing a proof.

1) What is/are given? Have I used anything that is given?

2) What do I want to show?

3) Is there anything you know that could help? Any theorems, propositions that are relevant?

Let me illustrate with the problem: Show that ker(A) is a subspace.

What is given? Ans: The definition of ker(A).

What do I want to show? Ans: ker(A) is a subspace.

Is there anything you know that could help? Ans: We know that in order to show that something is a subspace, we have to check the three criteria, namely:

1) That 0 \in ker(A).

2) That vector addition is closed.

3) That scalar multiplication is closed.

We proceed to verify the three criteria one by one:

1) We have to show that 0 \in ker(A).

How do you show that 0 \in ker(A)? We then resort to what is given, which is the defintion, and only the definition. We interpret the definition as “v \in ker(A) if Av = 0.” Therefore, in order to check that 0 \in ker(A), we only have to verify that A0 = 0, which is true.

In writing the proof, we don’t have to write everything down as I explained above, we can simply write the following:

“Since A0 = 0 , we have o \in ker(A).

Next, we move on to verify vector addition. We again ask ourselves the same set of questions:

What is given? Ans: u, v \in ker(A).

What does this mean? Ans: It means Au = 0 and Av = 0.

What do we want to show? Ans: We have to show that u + v \in ker(A).

How do we show this? Ans: It suffices to show A(u + v) = 0.

Okay, we then proceed to show A(u + v) = 0 using the fact that u, v \in ker(A).

Now, A(u + v) = Au + Av = 0 + 0 = 0 since u, v \in ker(A). Therefore, we have u + v \in ker(A).

Again, if we were to write down a proof, all we need to write down is the following:

Suppose u, v \in ker(A), we have Au = Av = 0. Hence we have A(u + v) = Au + Av = 0 + 0 = 0, and therefore u + v \in ker(A).

Okay, I think this is enough for problem 1a, and the verification of scalar multiplication will be left as an exercise.

For 1b, Sym_n(R) = \{ A \in M_n(R) \, | \, A^T = A \}. This means: A \in Sym_n(R) if A^T = A.

For 2, the dim (Sym_n(R)) = 1 + 2 + \dots + n = n(n + 1)/2, and my hint is to imitate what I did in the case n = 2 to find a pattern, and the same hint applies to finding a basis for n = 3 .

Problem 3

a) Show that any set containing n + 1 vectors in R^{n} is a linearly
dependent set.

(b) Explain without computation why the following determinant is
equal to 0:

\begin{pmatrix}1 & 2 & 3 & 4 & 5\\6 & 7 & 8 & 9 &10 \\11 & 12& 0 & 0 & 0\\13& 14& 0 & 0 & 0\\15& 16& 0 & 0 &0\end{pmatrix}.

This is really easy, answer the following questions for part (a).

What is the definition of dimension?

What is the dimension of R^n?

For part b, let U = \{ (x, y, 0, 0, 0)^T \, | \, x, y \in R\}. What is the dimension of U?

Problem 4

If A is a linear transformation on an $n$-dimensional vector space V and if A^{n -1}x \not = 0, but A^{n}x = 0, for some x \in V. Prove that x, Ax, \dots, A^{n-1}x are linearly independent. What are the eigenvalues of A.

In order to find the eigenvalues, the key observation is that x, Ax, \dots, A^{n-1}x forms a BASIS for V. The reason is that they are maximally linearly independent since the number of vectors matches the dimension of V.

Posted in Linear Algebra, Teaching |

Leave a Comment