Tutorial 3 - Linear Algebra - Vectors and Matrices.ipynb

Kernel: SageMath (stable)

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Linear Algebra

Vectors

To create a vector in Sage, use the vector command.

Exercise: Create the vector $x = (1, 2, \ldots, 100)$.

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Note: vectors in Sage are row vectors!

Exercise: Create the vector $y = (1^2, 2^2, \ldots, 100^2)$.

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<p><strong>Exercise: </strong>Type <strong>x.</strong> and hit tab to see the available methods for vectors. Find the <em>norm</em> (length) of the vectors <strong>x</strong> and <strong>y</strong>.</p>

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Exercise: Find the dot product of x and y.

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[The above problems are essentially the first problem on Exercise Set 1 of William Stein's Math 480b.]

Matrices

Exercise: Use the matrix command to create the following matrix over the rational numbers (hint: in Sage, QQ denotes the field of rational numbers).

$$\left(\begin{array}{rrrrrr}
3 & 2 & 2 & 1 & 1 & 0 \\
2 & 3 & 1 & 0 & 2 & 1 \\
2 & 1 & 3 & 2 & 0 & 1 \\
1 & 0 & 2 & 3 & 1 & 2 \\
1 & 2 & 0 & 1 & 3 & 2 \\
0 & 1 & 1 & 2 & 2 & 3
\end{array}\right)$$

1. Find the echelon form of the above matrix.
2. Find the right kernel of the matrix.
3. Find the eigenvalues of the matrix.
4. Find the left eigenvectors of the matrix.
5. Find the right eigenspaces of the matrix.

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Exercise: For what values of $k$ is the determinant of the following matrix $0$?

$$\left(\begin{array}{rrr}
1 & 1 & -1 \\
2 & 3 & k \\
1 & k & 3
\end{array}\right)$$

[K. R. Matthews, Elementary Linear Algebra, Chapter 4, Problem 8]

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Exercise: Prove that the determinant of the following matrix is $-8$.

$$\left(\begin{array}{rrr}
{n}^{2} & {\left( n + 1 \right)}^{2} & {\left( n + 2
\right)}^{2} \\
{\left( n + 1 \right)}^{2} & {\left( n + 2 \right)}^{2} &
{\left( n + 3 \right)}^{2} \\
{\left( n + 2 \right)}^{2} & {\left( n + 3 \right)}^{2} & {\left( n + 4 \right)}^{2}\end{array}\right)$$

[K. R. Matthews, Elementary Linear Algebra, Chapter 4, Problem 3]

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Exercise: Prove that if $a \neq c$, then the line through the points $(a,b)$ and $(c,d)$ is given by the following equation.

$$\det\left(\begin{array}{rrr}
x & y & 1 \\
a & b & 1 \\
c & d & 1
\end{array}\right) = 0.$$

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Exercise: Find the determinant of the following matrices.

$$\left(\begin{array}{r}
1
\end{array}\right),
\left(\begin{array}{rr}
1 & 1 \\
r & 1
\end{array}\right),
\left(\begin{array}{rrr}
1 & 1 & 1 \\
r & 1 & 1 \\
r & r & 1
\end{array}\right),
\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
r & 1 & 1 & 1 \\
r & r & 1 & 1 \\
r & r & r & 1
\end{array}\right),
\left(\begin{array}{rrrrr}
1 & 1 & 1 & 1 & 1 \\
r & 1 & 1 & 1 & 1 \\
r & r & 1 & 1 & 1 \\
r & r & r & 1 & 1 \\
r & r & r & r & 1
\end{array}\right)$$

Make a conjecture about the determinant of an arbitrary matrix in this sequence. Can you prove it your conjecture?

[Adapted from: K. R. Matthews, Elementary Linear Algebra, Chapter 4, Problem 19]

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Exercise: What is the largest determinant possible for a $3\times3$ matrix whose entries are $1, 2, \dots, 9$ (each occurring exactly once, in any order). How many matrices $M$ achieve this maximum?

(Hint: You might find the command Permutations useful. The following code will construct all the lists that have the entries $1, 2, 3, 4$, each appearing exactly once.)

for P in Permutations(4):
    L = list(P)
    print L

for P in Permutations(4):
    L = list(P)
    print L