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CPS 304 Homework 1 Due: Jan. 30

1)
Suppose you have the following vectors/matrices with indicated dimensions:

$\displaystyle A( = 3\times 5);\quad B(= 6 \times 10); \quad C(=2 \times 10); \quad
D(= 5 \times 5);$

$\displaystyle x(= 6 \times 1); \quad y(= 10 \times 1); \quad z(= 5 \times 1).$

What are the resulting dimensions of the following calculations:

$\displaystyle AD,\,\,DA,\,\,BC,\,\,BC^T,\,\,x^TB,\,\,By,\,\,ADz,\,\,x^TBy?$

If you cannot perform an operation, state why not. Also, if the result is a vector, state whether it is a row vector or column vector.

2)
Given the vectors

$\displaystyle x = \left(\begin{array}{r} 1 \\ 2 \end{array}\right),
\quad y = \left(\begin{array}{r} 3 \\ -1 \end{array}\right)$

compute the following: $ x - y,\,\, x+ 2y,\,\, x^T\,y,\,\, y^Tx.$
3)
Given the quantities

$\displaystyle A = \left(\begin{array}{rr} -3 & 1 \\ 0 & 4 \\ 1 & -2 \end{array}...
...ray}\right); \quad
y = \left(\begin{array}{r} 1 \\ -1 \\ -2 \end{array}\right)$

compute the following: $ A^T,\,\, Ax,\,\, A^Tx,\,\, y^TA,\,\, y^TB,\,\, AB,\,\, BA.$ If a quantity cannot be computed, state the reason why.
4)
For $ A$ and $ x$ as defined above, compute $ \Vert x \Vert _1,\,\, \Vert x \Vert _2,\,\, \Vert x \Vert _\infty,\,\, \Vert A \Vert _1,\,\, \Vert A \Vert _\infty,\,\,
\Vert A \Vert _F.$
5)
Prove that

$\displaystyle \Vert x \Vert _1 = \displaystyle{\sum_{i = 1}^{n} \vert x_i\vert}
$

is a norm (i.e., show that this definition satisfies conditions i) - iv) in Section 4 of the Linear Algebra Handout).
6)
Prove that

$\displaystyle \Vert x \Vert _2^2 = x^Tx.
$

7)
Prove that

$\displaystyle \lim_{p \to \infty} \Vert x \Vert _p = \Vert x \Vert _{\infty}.$

8)
(Extra Credit) Prove that

$\displaystyle \Vert A\Vert _{\mbox{max}} = \max_{\substack{i = 1,\ldots,n \\ j = 1,\ldots,m}} \vert a_{i,j} \vert $

is not a norm (HINT: Find 2 matrices that violate condition v) in Section 4).




2015-01-25