Chapter 2, Euclidean Space Video Solutions, Calculus and Analysis in Euclidean Space

Problem 1

Write down any three specific nonzero vectors $u, v, w$ from $\mathbb{R}^3$ and any two specific nonzero scalars $a, b$ from $\mathbb{R}$. Compute $u+v, a w, b(v+w),(a+b) u$, $u+v+w, a b w$, and the additive inverse to $u$.

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Problem 2

Working in $\mathbb{R}^2$, give a geometric proof that if we view the vectors $x$ and $y$ as arrows from 0 and form the parallelogram $P$ with these arrows as two of its sides, then the diagonal $z$ starting at $\mathbf{0}$ is the vector sum $x+y$ viewed as an arrow.

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17:34

Problem 3

Verify that $\mathbb{R}^n$ satisfies vector space axioms (A2), (A3), (D1).

Donald Albin

Numerade Educator

14:03

Problem 4

Are all the field axioms used in verifying that Euclidean space satisfies the vector space axioms?

Anthony Ramos

Numerade Educator

Problem 5

Show that $\mathbf{0}$ is the unique additive identity in $\mathbb{R}^n$. Show that each vector $x \in \mathbb{R}^n$ has a unique additive inverse, which can therefore be denoted $-x$. (And it follows that vector subtraction can now be defined,

$$
\left.-: \mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{R}^n, \quad x-y=x+(-y) \quad \text { for all } x, y \in \mathbb{R}^n .\right)
$$

Show that $0 x=0$ for all $x \in \mathbb{R}^n$.

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Problem 6

Repeat the previous exercise, but with $\mathbb{R}^n$ replaced by an arbitrary vector space $V$ over a field $F$. (Work with the axioms.)

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01:54

Problem 7

Show the uniqueness of the additive identity and the additive inverse using only (A1), (A2), (A3). (This is tricky; the opening pages of some books on group theory will help.)

Olivier Anderson

Numerade Educator

00:53

Problem 8

Let $x$ and $y$ be noncollinear vectors in $\mathbb{R}^3$. Give a geometric description of the set of all linear combinations of $x$ and $y$.

Victor Salazar

Numerade Educator

07:54

Problem 9

Which of the following sets are bases of $\mathbb{R}^3$ ?

$$
\begin{aligned}
& S_1=\{(1,0,0),(1,1,0),(1,1,1)\}, \\
& S_2=\{(1,0,0),(0,1,0),(0,0,1),(1,1,1)\}, \\
& S_3=\{(1,1,0),(0,1,1)\}, \\
& S_4=\{(1,1,0),(0,1,1),(1,0,-1)\} .
\end{aligned}
$$

How many elements do you think a basis for $\mathbb{R}^n$ must have? Give (without proof) geometric descriptions of all bases of $\mathbb{R}^2$, of $\mathbb{R}^3$.

Brian Ketelobeter

Numerade Educator

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Problem 10

Recall the field $\mathbb{C}$ of complex numbers. Define complex $n$-space $\mathbb{C}^n$ analogously to $\mathbb{R}^n$ :

$$
\mathbb{C}^n=\left\{\left(z_1, \ldots, z_n\right): z_i \in \mathbb{C} \text { for } i=1, \ldots, n\right\},
$$

and endow it with addition and scalar multiplication defined by the same formulas as for $\mathbb{R}^n$. You may take for granted that under these definitions, $\mathbb{C}^n$ satisfies the vector space axioms with scalar multiplication by scalars from $\mathbb{R}$, and also $\mathbb{C}^n$ satisfies the vector space axioms with scalar multiplication by scalars from $\mathbb{C}$. That is, using language that was introduced briefly in this section, $\mathbb{C}^n$ can be viewed as a vector space over $\mathbb{R}$ and also, separately, as a vector space over $\mathbb{C}$. Give a basis for each of these vector spaces.

Andrew Eddins

Emory University