Problem # X.1:
(a) Check that the vectors \(\mathbf{v}=(1,2,1)\)
and \(\mathbf{w}=(8,4,-1)\) are solutions to the linear equation
\[
2x_1 - 3x_2 + 4x_3 = 0. \qquad\text{(1)}
\]
Problem # X.2:
An \(m\)-by-\(n\) matrix with coefficients in a field \(\mathbb{F}\)
is defined to be an \(m\)-by-\(n\) array of elements of \(\mathbb{F}\). We
write \(M_{m,n}(\mathbb{F})\) for the set of all such matrices, so an
element \(A\in M_{m,n}(\mathbb{F})\) looks like
\[
A = \begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{pmatrix}
\]
We make \(M_{m,n}(\mathbb{F})\) into a vector space in
the obvious way:
\[
\begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & a_{mn} \\
\end{pmatrix}
+
\begin{pmatrix}
b_{11} & \cdots & b_{1n} \\
\vdots & \ddots & \vdots \\
b_{m1} & \cdots & b_{mn} \\
\end{pmatrix}
=
\begin{pmatrix}
a_{11}+b_{11} & \cdots & a_{1n}+b_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1}+b_{m1} & \cdots & a_{mn}+b_{mn} \\
\end{pmatrix}
\]
and
\[
c \begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & a_{mn} \\
\end{pmatrix}
=
\begin{pmatrix}
ca_{11} & \cdots & ca_{1n} \\
\vdots & \ddots & \vdots \\
ca_{m1} & \cdots & ca_{mn} \\
\end{pmatrix}
\]
Problem # X.3:
The transpose of a (square)
matrix \(A\), denoted \(A^*\), is obtained
by flipping the entries across the main diagonal. So for example
\[
\begin{pmatrix} 1&2&3\\ 4&5&6\\ 7&8&9\\ \end{pmatrix}^*
=
\begin{pmatrix} 1&4&7\\ 2&5&8\\ 3&6&9\\ \end{pmatrix}.
\]
A matrix \(A\) is symmetric if \(A^*=A\)
and it is anti-symmetric if \(A^*=-A\).
Problem # X.4:
Find the matrix associated to each of the following
linear transformations relative to the given bases.
Problem # X.5:
Compute each of the following matrix products, or explain
why the product is not well-defined.
\[
\begin{aligned}
\hbox{(a)}\quad&
\begin{pmatrix}
1&2&3\\ 4&5&6\\
\end{pmatrix}
\begin{pmatrix}
3\\-1\\2\\
\end{pmatrix}
&
\quad\hbox{(b)}\quad&
\begin{pmatrix}
1&2&3\\ 4&5&6\\
\end{pmatrix}
\begin{pmatrix}
3\\2\\
\end{pmatrix}
&
\quad\hbox{(c)}\quad&
\begin{pmatrix}
1&-1&2\\ 2&1&3\\ -4&2&5\\
\end{pmatrix}
\begin{pmatrix}
2&1\\ -1&3\\ 4&-2\\
\end{pmatrix}
\\
\hbox{(d)}\quad&
\begin{pmatrix}
1&2&3&4\\
\end{pmatrix}
\begin{pmatrix}
1\\2\\3\\4\\
\end{pmatrix}
&
\quad\hbox{(e)}\quad&
\begin{pmatrix}
1\\2\\3\\4\\
\end{pmatrix}
\begin{pmatrix}
1&2&3&4\\
\end{pmatrix}
&
\quad\hbox{(f)}\quad&
\begin{pmatrix}
3&1&0&0\\ 0&3&1&0\\ 0&0&3&1\\ 0&0&0&3\\
\end{pmatrix}
\begin{pmatrix}
3&1&0&0\\ 0&3&1&0\\ 0&0&3&1\\ 0&0&0&3\\
\end{pmatrix}
\\
\end{aligned}
\]
Problem # X.6:
(a) Find linear transformations (or matrices)
\(A,B:\mathbb{F}^2\to\mathbb{F}^2\) with the property
that \(AB=0\), but \(BA\ne0\).
Problem # X.7:
(a) Give a sequence of swaps that changes the sequence (6,3,1,2,5,4)
into the sequence (1,2,3,4,5,6). Use you computation to
compute the sign of the permutation (6,3,1,2,5,4).
Problem # X.8:
Apply the Gram–Schmidt algorithm to the three vectors
\[
(1,2,2),\quad (1,1,1)\quad (3,2,1)
\quad\text{in}\quad \mathbb{R}^3
\]
to create an orthonormal basis. (You can either just keep track
of quantities to 4 or 5 decimal places, or express your answer exactly
using square roots.)
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(b) The vector \(\mathbf{u}=(-37,-14,8)\) is also a solution to equation (1).
Find real numbers \(a,b\in\mathbf{R}\) so that
\[
\mathbf{u} = a\mathbf{v}+b\mathbf{w}.
\]
(c) Prove the following general result.
If the vector \(\mathbf{z}=(z_1,z_2,z_3)\in\mathbf{R}^3\) is
a solution to equation (1), then there are scalars
\(a,b\in\mathbf{R}\) so that
\[
\mathbf{z} = a\mathbf{v}+b\mathbf{w}. \qquad\text{(2)}
\]
(d) In (c), prove that for a given vector \(\mathbf{z}\),
there is only one choice for \(a\) and \(b\) that makes equation (2) true.
(a)
Write down a basis for the \(\mathbb{F}\)-vector space
\(M_{3,2}(\mathbb{F})\) of 3-by-2 matrices. What is the dimension of
\(M_{3,2}(\mathbb{F})\)?
(b)
More generally, what is the dimension of
\(M_{m,n}(\mathbb{F})\)?
(a)
Prove that the set of \(n\)-by-\(n\) symmetric matrices is a
vector subspace of \(M_{n,n}(\mathbb{F})\).
(b)
Find a basis for the space of 2-by-2 symmetric matrices. What
is its dimension?
(c)
Generalize by describing a basis for the space of \(n\)-by-\(n\) symmetric
matrices and computing its dimension. (It might help to start with 3-by-3.)
(d)
Prove that the set of
\(n\)-by-\(n\) anti-symmetric matrices is also a vector subspace
of \(M_{n,n}(\mathbb{F})\),
describe a basis, and compute its dimension.
(e) (Bonus)
Let's write \(M_{n,n}(\mathbb{F})^{\text{sym}}\)
for the space of symmetric matrices and
\(M_{n,n}(\mathbb{F})^{\text{anti-sym}}\)
for the space of anti-symmetric matrices. Prove that
\[
M_{n,n}(\mathbb{F})
= M_{n,n}(\mathbb{F})^{\text{sym}} + M_{n,n}(\mathbb{F})^{\text{anti-sym}}.
\]
Is this a direct sum of vector spaces?
(a)
\(T:\mathbb{F}^3\to\mathbb{F}^3\) defined by
\(T(x,y,z)=(2x-3y,3x+5z,7x+2y-z)\),
using the standard basis for \(\mathbb{F}^3\).
(b)
\(T:\mathbb{F}^2\to\mathbb{F}^3\) defined by
\(T(x,y,z)=(2x-3y,3x+5y,-x)\),
using the standard bases for \(\mathbb{F}^2\) and \(\mathbb{F}^3\).
(c)
Let \(\mathcal{P}_n(\mathbb{F})\) be the \(\mathbb{F}\)-vector space
of polynomials of degree at most \(n\) with coefficients in \(\mathbb{F}\).
Use the basis \(\{1,t,t^2,\ldots,t^n\}\). What is the matrix associated
to the linear transformation
\(T:\mathcal{P}_n(\mathbb{F})\to\mathcal{P}_n(\mathbb{F})\) defined
by differentiation \(T(f(t)) = f'(t)\)?
[Hint. If you don't see immediately how to do it, try
doing \(n=2\) and \(n=3\) to get an idea what's going on.]
(d)
Continuing with the notation from (c), what is the matrix
for the linear transformation
\(T:\mathcal{P}_n(\mathbb{F})\to\mathcal{P}_{n+2}(\mathbb{F})\) defined
\(T(f(t))=(t^2+1)f(t)\)?
(e)
Let \(V\) be the real vector space consisting of all functions of
the form \(f(t)=ae^{2t}+bte^{2t}\) with \(a\) and \(b\) in \(\mathbb{R}\).
Using the basis \(\{e^{2t},te^{2t}\}\), what is the matrix associated
to the linear transformation \(T:V\to V\) defined
by \(T(f(t))=f'(t)\)?
(b)
Find a linear transformation (or matrix)
\(A:\mathbb{F}^2\to\mathbb{F}^2\) with the property
that \(A\ne 0\), but \(A^2=0\).
(b) The sequence (4,3,2,1) can be put into order by the
following eight swaps:
\[
(4,3,2,1) \to (4,3,1,2) \to (4,1,3,2) \to (4,1,2,3)
\to (4,2,1,3) \to (2,4,1,3) \to (2,1,4,3) \to (1,2,4,3) \to (1,2,3,4).
\]
Find a different way to put the sequence in order using
only six swaps. What is the sign of the permuation (4,3,2,1)?