Draw the following school supplies. cuaderno lápiz mochila bolígrafo

How is the one's complement representation of the difference of two integers obtained from the one's complement representations of those integers?

Aquí está tu cuaderno. - ¿Dónde?

Find the values of the following products. \text { a) $\prod_{j=2}^{n}(1-1 / j)$ b) $\prod_{j=2}^{n}\left(1-1 / j^{2}\right)$

Find all positive integers that divide each of these integers. a) 12, b) 22, c) 37, d) 41.

Prove that the space of all $m \times n$ matrices over the field F has dimension mn, by exhibiting a basis for this space.

Let T be a linear operator on V. Suppose $V=W_{1} \oplus \cdots \oplus W_{k}$, where each $W_i$ is invariant under T. Let $T_i$ be the induced (restriction) operator on $W_i$. (a) Prove that det $(T)=\operatorname{det}\left(T_{1}\right) \cdots \operatorname{det}\left(T_{k}\right)$. (b) Prove that the characteristic polynomial for f is the product of the charaeteristic polynomials for $f_{1}, \dots, f_{k}$. (c) Prove that the minimal polynomial for T is the least common multiple of the minimal polynomials for $T_{1}, \ldots, T_{k}$.

Label each sentence dec. if it is declarative and imp. if it is imperative. Louie XIV was known as the "Sun King."

Label each sentence dec. if it is declarative and imp. if it is imperative. We're sailing from Cleveland to Erie Beach, Ontario.

Let n be a positive integer and F a field. Let W be the set of all vectors $\left(x_{1}, \dots, x_{n}\right)$ in $F^{n}$ such that $x_{1}+\cdots+x_{n}=0$. (a) Prove that $W^0$ consists of all linear functionals f of the form $f\left(x_{1}, \dots, x_{n}\right)=c \sum_{j=1}^{n} x_{j}$. (b) Show that the dual space W’* of W can be ‘naturally’ identified with the linear functionals $f\left(x_{1}, \ldots, x_{n}\right)=c_{1} x_{1}+\cdots+c_{n} x_{n}$ on $F^{n}$ which satisfy $c_{1}+\cdots+c_{n}=0$.

The following expressions define quadratic forms q on $R^2$. Find the symmetric bilinear form f corresponding to each q. (a) $a x_{1}^{2}$ (b) $b x_{1} x_{2}$ (c) $c x_{2}^{2}$ (d) $2 x_{1}^{2}-\frac{1}{3} x_{1} x_{2}$ (e) $x_{1}^{2}+9 x_{2}^{2}$ (f) $3 x_{1} x_{2}-x_{2}^{2}$ (g) $4 x_{1}^{2}+6 x_{1} x_{2}-3 x_{2}^{2}$

Let F be a field and let n be a positive integer $(n \geq 2)$. Let V be the vector space of all $n \times n$ matrices over F. Which of the following sets of matrices B in V are subspaces of V? (a) all invertible A ; (b) all non-invertible A ; (c) all A such that AB = BA, where B is some fixed matrix in V; (d) all A such that $A^{2}=A$.

Prove that the determinant of the Vandermonde matrix

$$\left[\begin{array}{lll}{1} & {a} & {a^{2}} \\ {1} & {b} & {b^{2}} \\ {1} & {c} & {c^{2}}\end{array}\right]$$

is (b - a)(c - a)(c - b).

Find each of the following sums. a) $\sum_{j=0}^{4} 3$, b) $\sum_{j=0}^{4}(j-3)$, c) $\sum_{j=0}^{4}(j+1) /(j+2)$.