Preparing for the Exam (2)

Here are a collection of relatively easy problems:

1) Can you find a 4 by 4 matrix, that is NOT upper-triangular with charpoly being

2) When you have a matrix, always ask yourself the following questions:

Answer to Basic Linear Algebra Questions (1)

The following is a list of basic questions you should know how to answer (by heart)

(1) Suppose are linearly independent and
What do you know about the

Ans: for all

(2) If , what is

Ans: Ax = 0.

Hints to Selected Problems in HW 5

For 3. Answer the following mini questions:
(a) Let A = B + I, what is rank(A)?
(b) What is
(c) If , what should be in

Preparing for the Linear Algebra Final (1)

The following is a list of basic questions you should know how to answer (by heart)

(1) Suppose are linearly independent and
What do you know about the

(2) If , what is

Magic Squares (Proof)

In this post, we will prove two things:

Let be the space of by magic squares.

1) That if , then for
2) I will find the dimension, as well as a basis for

For 1, Let , and we first assume

Hints to Selected Problems in HW 4

1b: Use part (a), Proposition 4.41 and rank-nullity.

2c: Use 2b, and HW problem 3a.

2e: what are the polynomials such that f’ = 0 ?
2f: Use rank-nullity.
2g: What is
2l: First find

The problems are available at:
Click here

Questions concerning Magic Squares and Matrices

A magic square is such that all row sums, column sums, and diagonal sums are equal, and we call this common sum the “magic number” for the magic square.  Let’s look at the following well-known 3 by 3 magic square with magic number 15:

If we view this as a 3 by 3 matrix, what kind of properties does possess?

Here is what I found out:

Hints to Selected Problems in HW 3

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 This literally translate into:

is the set of vectors in such that Basically, it tells you a criterion to test whether a vector is in , and you can also interpret it as