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ESC 251 Homework 2 Due: Sep. 5

1)
(2 pts) 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)
(4 pts) 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)
(11 pts) 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)
(6 pts) For $ A$ and $ x$ as defined in Problem 4, 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)
(2 pts) Prove that

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





2018-08-31