Vector Spaces

Show that $\alpha +\beta = \beta +\alpha$ for all $\alpha , \beta \in \mathbb {C}$.

Show that $(\alpha +\beta )+\lambda = \alpha +(\beta +\lambda )$ for all $\alpha , \beta , \lambda \in \mathbb {C}$.

Show that $(\alpha \beta ) \lambda = \alpha (\beta \lambda )$ for all $\alpha , \beta , \lambda \in \mathbb {C}$.

Show that $\lambda (\alpha +\beta ) = \lambda \alpha +\lambda \beta$ for all $\lambda , \alpha , \beta \in \mathbb {C}$.

Show that for every $\alpha \in \mathbb {C}$, there exists a unique $\beta \in \mathbb {C}$ such that $\alpha +\beta = 0$.

Show that for every $\alpha \in \mathbb {C}$ with $\alpha \neq 0$, there exists a unique $\beta \in \mathbb {C}$ such that $\alpha \beta = 1$.

Show that

is a cube root of 1 (meaning that its cube equals 1).

Find two distinct square roots of $i$.

Find $x \in \mathbb {R}^{4}$ such that

Explain why there does not exist $\lambda \in \mathbb {C}$ such that

Show that $(x+y)+z = x+(y+z)$ for all $x, y, z \in \mathbf{F}^{n}$.

Show that $(a b) x = a(b x)$ for all $x \in \mathbf{F}^{n}$ and all $a, b \in \mathbf{F}$.

Show that $1 x = x$ for all $x \in \mathbf{F}^{n}$.

Show that $\lambda (x+y) = \lambda x+\lambda y$ for all $\lambda \in \mathbf{F}$ and all $x, y \in \mathbf{F}^{n}$.

Show that $(a+b) x = a x+b x$ for all $a, b \in \mathbf{F}$ and all $x \in \mathbf{F}^{n}$.

Prove that $-(-v) = v$ for every $v \in V$.

Suppose $a \in \mathbf{F}, v \in V$, and $a v = 0$. Prove that $a = 0$ or $v = 0$.

Suppose $v, w \in V$. Explain why there exists a unique $x \in V$ such that $v+3 x = w$.

The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in the definition of a vector space (1.20). Which one?

Definition 1.20 (vector space): a vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on $V$ such that the following properties hold. Commutativity: $u+v = v+u$ for all $u, v \in V$. Associativity: $(u+v)+w = u+(v+w)$ and $(a b) v = a(b v)$ for all $u, v, w \in V$ and for all $a, b \in \mathbf{F}$. Additive identity: there exists an element $0 \in V$ such that $v+0 = v$ for all $v \in V$. Additive inverse: for every $v \in V$, there exists $w \in V$ such that $v+w = 0$. Multiplicative identity: $1 v = v$ for all $v \in V$. Distributive properties: $a(u+v) = a u+a v$ and $(a+b) v = a v+b v$ for all $a, b \in \mathbf{F}$ and all $u, v \in V$.

Show that in the definition of a vector space (1.20), the additive inverse condition can be replaced with the condition that

Here the 0 on the left side is the number 0, and the 0 on the right side is the additive identity of $V$.

Definition 1.20 (vector space): a vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on $V$ such that the following properties hold. Commutativity: $u+v = v+u$ for all $u, v \in V$. Associativity: $(u+v)+w = u+(v+w)$ and $(a b) v = a(b v)$ for all $u, v, w \in V$ and for all $a, b \in \mathbf{F}$. Additive identity: there exists an element $0 \in V$ such that $v+0 = v$ for all $v \in V$. Additive inverse: for every $v \in V$, there exists $w \in V$ such that $v+w = 0$. Multiplicative identity: $1 v = v$ for all $v \in V$. Distributive properties: $a(u+v) = a u+a v$ and $(a+b) v = a v+b v$ for all $a, b \in \mathbf{F}$ and all $u, v \in V$.

Let $\infty$ and $-\infty$ denote two distinct objects, neither of which is in $\mathbb {R}$. Define an addition and scalar multiplication on $\mathbb {R} \cup \left\{ \infty ,-\infty \right\}$ as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for $t \in \mathbb {R}$ define

and

With these operations of addition and scalar multiplication, is $\mathbb {R} \cup \left\{ \infty ,-\infty \right\}$ a vector space over $\mathbb {R}$? Explain.

Suppose $S$ is a nonempty set. Let $V^{S}$ denote the set of functions from $S$ to $V$. Define a natural addition and scalar multiplication on $V^{S}$, and show that $V^{S}$ is a vector space with these definitions.

Suppose $V$ is a real vector space.

• The complexification of $V$, denoted by $V_{\mathrm{C}}$, equals $V \times V$. An element of $V_{\mathrm{C}}$ is an ordered pair $(u, v)$, where $u, v \in V$, but we write this as $u+i v$.

• Addition on $V_{\mathrm{C}}$ is defined by

  $$\left(u_{1}+i v_{1}\right)+\left(u_{2}+i v_{2}\right) = \left(u_{1}+u_{2}\right)+i\left(v_{1}+v_{2}\right)$$

  for all $u_{1}, v_{1}, u_{2}, v_{2} \in V$.

• Complex scalar multiplication on $V_{\mathrm{C}}$ is defined by

  $$(a+b i)(u+i v) = (a u-b v)+i(a v+b u)$$

  for all $a, b \in \mathbb {R}$ and all $u, v \in V$.

Prove that with the definitions of addition and scalar multiplication as above, $V_{\mathrm{C}}$ is a complex vector space.

For each of the following subsets of $\mathbf{F}^{3}$, determine whether it is a subspace of $\mathbf{F}^{3}$.

Verify all assertions about subspaces in Example 1.35.

Example 1.35:

Show that the set of differentiable real-valued functions $f$ on the interval $(-4,4)$ such that $f^{\prime }(-1) = 3 f(2)$ is a subspace of $\mathbb {R}^{(-4,4)}$.

Suppose $b \in \mathbb {R}$. Show that the set of continuous real-valued functions $f$ on the interval $[0,1]$ such that $\int_{0}^{1} f = b$ is a subspace of $\mathbb {R}^{[0,1]}$ if and only if $b = 0$.

Is $\mathbb {R}^{2}$ a subspace of the complex vector space $\mathbb {C}^{2}$?

Prove or give a counterexample: If $U$ is a nonempty subset of $\mathbb {R}^{2}$ such that $U$ is closed under addition and under taking additive inverses (meaning $-u \in U$ whenever $u \in U$), then $U$ is a subspace of $\mathbb {R}^{2}$.

Give an example of a nonempty subset $U$ of $\mathbb {R}^{2}$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\mathbb {R}^{2}$.

A function $f: \mathbb {R} \rightarrow \mathbb {R}$ is called periodic if there exists a positive number $p$ such that $f(x) = f(x+p)$ for all $x \in \mathbb {R}$. Is the set of periodic functions from $\mathbb {R}$ to $\mathbb {R}$ a subspace of $\mathbb {R}^{\mathbb {R}}$? Explain.

Suppose $V_{1}$ and $V_{2}$ are subspaces of $V$. Prove that the intersection $V_{1} \cap V_{2}$ is a subspace of $V$.

Prove that the intersection of every collection of subspaces of $V$ is a subspace of $V$.

Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces is contained in the other.

Prove that the union of three subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces contains the other two.

Suppose

Describe $U+W$ using symbols, and also give a description of $U+W$ that uses no symbols.

Suppose $U$ is a subspace of $V$. What is $U+U$?

Is the operation of addition on the subspaces of $V$ commutative? In other words, if $U$ and $W$ are subspaces of $V$, is $U+W = W+U$?

Is the operation of addition on the subspaces of $V$ associative? In other words, if $V_{1}, V_{2}, V_{3}$ are subspaces of $V$, is

Does the operation of addition on the subspaces of $V$ have an additive identity? Which subspaces have additive inverses?

Prove or give a counterexample: If $V_{1}, V_{2}, U$ are subspaces of $V$ such that

then $V_{1} = V_{2}$.

Suppose

Find a subspace $W$ of $\mathbf{F}^{4}$ such that $\mathbf{F}^{4} = U \oplus W$.

Suppose

Find a subspace $W$ of $\mathbf{F}^{5}$ such that $\mathbf{F}^{5} = U \oplus W$.

Suppose

Find three subspaces $W_{1}, W_{2}, W_{3}$ of $\mathbf{F}^{5}$, none of which equals $\left\{ 0\right\}$, such that $\mathbf{F}^{5} = U \oplus W_{1} \oplus W_{2} \oplus W_{3}$.

Prove or give a counterexample: If $V_{1}, V_{2}, U$ are subspaces of $V$ such that

then $V_{1} = V_{2}$.

A function $f: \mathbb {R} \rightarrow \mathbb {R}$ is called even if

for all $x \in \mathbb {R}$. A function $f: \mathbb {R} \rightarrow \mathbb {R}$ is called odd if

for all $x \in \mathbb {R}$. Let $V_{\mathrm{e}}$ denote the set of real-valued even functions on $\mathbb {R}$ and let $V_{\mathrm{o}}$ denote the set of real-valued odd functions on $\mathbb {R}$. Show that $\mathbb {R}^{\mathbb {R}} = V_{\mathrm{e}} \oplus V_{\mathrm{o}}$.