Assignment
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Assigned
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Due
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Problems
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| HW1 |
1/23/14
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1/29/14
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1. Do Exercise 1.3.2.
2. Do Exercise 2.3.2. Explain why your answer is correct.
3. Do Exercise 3.2.1.
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| HW2 |
1/31/14
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2/5/14
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1. Do Exercise 2.5.7.
2. Do Exercise 3.5.5.
3. Do Exercise 3.7.2.
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| HW3 |
2/6/14
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2/14/14
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1.
(a) Find a right inverse of the matrix
$$
\left[
\begin{array}{ccc}
1&2&3\\
4&5&6
\end{array}
\right]
$$
or explain why none exists.
(b) Find a left inverse of
$$
\left[
\begin{array}{cc}
1&2\\
2&4\\
3&6
\end{array}
\right]
$$
or explain why none exists.
2.
(a) Show that the set of functions $\{x,x+1,x+2\}$ is linearly
dependent.
(b) Show that the set of functions $\{1,\sin^2(x),\cos^2(x)\}$ is linearly
dependent.
3. For $A\in M_{m\times n}(\mathbb R)$, show that the
solution set of $A{\bf x}={\bf 0}$ is a subspace
of $\mathbb R^n$.
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| HW4 |
2/15/14
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2/21/14
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1. Do Exercise 4.1.1 parts (a), (b), (c). (For each part,
say whether the set is a subspace, then give a very
brief justification for your answer.)
2. Do Exercises 4.3.2 and 4.3.7.
3. Do Exercise 4.4.1.
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| HW5 |
2/19/14
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2/26/14
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1. Do Exercise 4.2.1, except replace the phrase "spanning sets"
with "bases".
2. Do Exercises 4.2.3 and 4.2.5.
3. Do Exercise 4.2.6(a)(b).
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| HW6 |
3/5/14
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3/12/14
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1. Do Exercise 4.7.8.
2. Do Exercises 4.7.12(a) and 4.7.13.
3. Do Exercise 4.7.18.
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| HW7 |
3/12/14
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3/19/14
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1. Do Exercise 4.8.1.
2. Do Exercises 4.8.2 and 4.8.9.
3. Do Exercises 4.8.3 and 4.8.4. (Note: for 4.8.3(b), the matrix
$Q$ should be ${}_{\mathcal S}[\textrm{id}]_{{\mathcal S}'}$.)
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| HW8 |
3/19/14
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4/2/14
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1. Do this network flow problem.
2. Balance the equation $B_2S_3+H_2O\to H_3BO_3+H_2S$.
3. Consider the production model ${\bf x}=C{\bf x}+{\bf d}$
for an economy with two sectors where
$C=\left[\begin{array}{cc}.2&.5\\.6&.1\end{array}\right]$ and
${\bf d}=\left[\begin{array}{cc}16\\12\end{array}\right]$.
Solve for the equilibirum production level.
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| HW9 |
4/3/14
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4/9/14
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1. Do Exercise 5.1.2(a)(b). Also, determine
(c) the unit vectors in the directions of ${\bf u}$ and ${\bf v}$, and
(d) the angle between ${\bf u}$ and ${\bf v}$.
2.
(a) Find a vector that is orthogonal to both
${\bf u}=\left[\begin{array}{c}0\\1\\1\end{array}\right]$ and
${\bf v}=\left[\begin{array}{c}1\\0\\1\end{array}\right]$.
(b) The points
$\left[\begin{array}{c}0\\0\\0\end{array}\right]$,
$\left[\begin{array}{c}0\\1\\1\end{array}\right]$,
$\left[\begin{array}{c}1\\0\\1\end{array}\right]$ and
$\left[\begin{array}{c}1\\1\\0\end{array}\right]$ are the vertices
of a regular tetrahedron. Find vectors orthogonal to
two adjacent faces.
(c) Find the angle between two adjacent faces of a regular tetrahedron.
3. Let $V = \mathbb P_2(t)$. Explain why the function
$$
\langle f,g\rangle:=\int_0^1 f(t)g(t)\;dt$$ is an inner product on $V$.
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| HW10 |
4/9/14
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4/16/14
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1. Let $V = M_{m\times n}(\mathbb C)$.
Explain why the function
$$
\langle A,B\rangle
= \textrm{trace}(A^H\cdot B)
= \textrm{trace}(A^*\cdot B)
$$
is a complex inner product on $V$.
2. Do Exercise 5.4.1(a)(b)(c)(d)
3. Let $A$ be an $m\times n$ real matrix.
Show that the null space of $A$ is the orthogonal
complement of the row space, and the left null space
is the orthogonal complement of the column space.
How are the dimensions of all of these spaces related?
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HW11
Last one!
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4/18/14
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4/23/14
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1. Do Exercise 5.5.2.
2. Do Exercise 5.6.8.
3. Do Exercise 5.13.12.
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