Definition and Examples

Name the zero vector for each of these vector spaces.

• The space of degree three polynomials under the natural operations.

• The space of $2\!\times\!4$ matrices.

• The space $\{\hskip 1.00006pt\text{$f\colon[0..1]\to\mathbb{R}$}\mid f\text{ is continuous}\hskip 1.00006pt\}$.

• The space of real-valued functions of one natural number variable.

Find the additive inverse, in the vector space, of the vector.

• In $\mathcal{P}_{3}$, the vector $-3-2x+x^{2}$.

• In the space $2\!\times\!2$,

• In $\{\hskip 1.00006ptae^{x}+be^{-x}\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$, the space of functions of the real variable $x$ under the natural operations, the vector $3e^{x}-2e^{-x}$.

For each, list three elements and then show it is a vector space.

• The set of linear polynomials $\mathcal{P}_{1}=\text{$\{\hskip 1.00006pta_{0}+a_{1}x\mid a_{0},a_{1}\in\mathbb{R}\hskip 1.00006pt\}$}$ under the usual polynomial addition and scalar multiplication operations.

• The set of linear polynomials $\{\hskip 1.00006pta_{0}+a_{1}x\mid a_{0}-2a_{1}=0\hskip 1.00006pt\}$, under the usual polynomial addition and scalar multiplication operations.

Hint. Use Example 1.3 as a guide. Most of the ten conditions are just verifications.

For each, list three elements and then show it is a vector space.

• The set of $2\!\times\!2$ matrices with real entries under the usual matrix operations.

• The set of $2\!\times\!2$ matrices with real entries where the $2,1$ entry is zero, under the usual matrix operations.

For each, list three elements and then show it is a vector space.

• The set of three-component row vectors with their usual operations.

• The set

  $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\in\mathbb{R}^{4}\mid x+y-z+w=0\hskip 1.00006pt\}$

  under the operations inherited from $\mathbb{R}^{4}$.

Show that each of these is not a vector space. (Hint. Check closure by listing two members of each set and trying some operations on them.)

• Under the operations inherited from $\mathbb{R}^{3}$, this set

  $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\in\mathbb{R}^{3}\mid x+y+z=1\hskip 1.00006pt\}$

• Under the operations inherited from $\mathbb{R}^{3}$, this set

  $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\in\mathbb{R}^{3}\mid x^{2}+y^{2}+z^{2}=1\hskip 1.00006pt\}$

• Under the usual matrix operations,

  $\{\hskip 1.00006pt\begin{pmatrix}a&1\\ b&c\end{pmatrix}\mid a,b,c\in\mathbb{R}\hskip 1.00006pt\}$

• Under the usual polynomial operations,

  $\{\hskip 1.00006pta_{0}+a_{1}x+a_{2}x^{2}\mid a_{0},a_{1},a_{2}\in\mathbb{R}^{+}\hskip 1.00006pt\}$

  where $\mathbb{R}^{+}$ is the set of reals greater than zero

• Under the inherited operat…

Define addition and scalar multiplication operations to make the complex numbers a vector space over $\mathbb{R}$.

Is the set of rational numbers a vector space over $\mathbb{R}$ under the usual addition and scalar multiplication operations?

Show that the set of linear combinations of the variables $x,y,z$ is a vector space under the natural addition and scalar multiplication operations.

Prove that this is not a vector space: the set of two-tall column vectors with real entries subject to these operations.

Prove or disprove that $\mathbb{R}^{3}$ is a vector space under these operations.

• $\begin{pmatrix}x_{1}\\ y_{1}\\ z_{1}\end{pmatrix}+\begin{pmatrix}x_{2}\\ y_{2}\\ z_{2}\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}\quad\text{and}\quad r\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}rx\\ ry\\ rz\end{pmatrix}$

• $\begin{pmatrix}x_{1}\\ y_{1}\\ z_{1}\end{pmatrix}+\begin{pmatrix}x_{2}\\ y_{2}\\ z_{2}\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}\quad\text{and}\quad r\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}$

For each, decide if it is a vector space; the intended operations are the natural ones.

• The diagonal $2\!\times\!2$ matrices

  $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ 0&b\end{pmatrix}\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$

• This set of $2\!\times\!2$ matrices

  $\{\hskip 1.00006pt\begin{pmatrix}x&x+y\\ x+y&y\end{pmatrix}\mid x,y\in\mathbb{R}\hskip 1.00006pt\}$

• This set

  $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\in\mathbb{R}^{4}\mid x+y+z+w=1\hskip 1.00006pt\}$

• The set of functions $\{\hskip 1.00006pt\text{$f\colon\mathbb{R}\to\mathbb{R}$}\mid df/dx+2f=0\hskip 1.00006pt\}$

• The set of functions $\{\hskip 1.00006pt\text{$f\colon\mathbb{R}\to\mathbb{R}$}\mid df/dx+2f=1\hskip 1.00006pt\}$

Prove or disprove that this is a vector space: the real-valued functions $f$ of one real variable such that $f(7)=0$.

Show that the set $\mathbb{R}^{+}$ of positive reals is a vector space when we interpret ‘$x+y$’ to mean the product of $x$ and $y$ (so that $2+3$ is $6$), and we interpret ‘$r\cdot x$’ as the $r$-th power of $x$.

Is $\{\hskip 1.00006pt(x,y)\mid x,y\in\mathbb{R}\hskip 1.00006pt\}$ a vector space under these operations?

• $(x_{1},y_{1})+(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2})$ and $r\cdot(x,y)=(rx,y)$

• $(x_{1},y_{1})+(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2})$ and $r\cdot(x,y)=(rx,0)$

Prove or disprove that this is a vector space: the set of polynomials of degree greater than or equal to two, along with the zero polynomial.

At this point “the same” is only an intuition, but nonetheless for each vector space identify the $k$ for which the space is “the same” as $\mathbb{R}^{k}$.

• The $2\!\times\!3$ matrices under the usual operations

• The $n\!\times\!m$ matrices (under their usual operations)

• This set of $2\!\times\!2$ matrices

  $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ b&c\end{pmatrix}\mid a,b,c\in\mathbb{R}\hskip 1.00006pt\}$

• This set of $2\!\times\!2$ matrices

  $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ b&c\end{pmatrix}\mid a+b+c=0\hskip 1.00006pt\}$

Using $\vec{+}$ to represent vector addition and $\,\vec{\cdot}\,$ for scalar multiplication, restate the definition of vector space.

Prove these.

• For any $\vec{v}\in V$, if $\vec{w}\in V$ is an additive inverse of $\vec{v}$, then $\vec{v}$ is an additive inverse of $\vec{w}$. So a vector is an additive inverse of any additive inverse of itself.

• Vector addition left-cancels: if $\vec{v},\vec{s},\vec{t}\in V$ then $\vec{v}+\vec{s}=\vec{v}+\vec{t}\,$ implies that $\vec{s}=\vec{t}$.

The definition of vector spaces does not explicitly say that $\vec{0}+\vec{v}=\vec{v}$ (it instead says that $\vec{v}+\vec{0}=\vec{v}$). Show that it must nonetheless hold in any vector space.

Prove or disprove that this is a vector space: the set of all matrices, under the usual operations.

In a vector space every element has an additive inverse. Can some elements have two or more?

• Prove that every point, line, or plane through the origin in $\mathbb{R}^{3}$ is a vector space under the inherited operations.

• What if it doesn’t contain the origin?

Using the idea of a vector space we can easily reprove that the solution set of a homogeneous linear system has either one element or infinitely many elements. Assume that $\vec{v}\in V$ is not $\vec{0}$.

• Prove that $r\cdot\vec{v}=\vec{0}$ if and only if $r=0$.

• Prove that $r_{1}\cdot\vec{v}=r_{2}\cdot\vec{v}$ if and only if $r_{1}=r_{2}$.

• Prove that any nontrivial vector space is infinite.

• Use the fact that a nonempty solution set of a homogeneous linear system is a vector space to…

Is this a vector space under the natural operations: the real-valued functions of one real variable that are differentiable?

A vector space over the complex numbers $\mathbb{C}$ has the same definition as a vector space over the reals except that scalars are drawn from $\mathbb{C}$ instead of from $\mathbb{R}$. Show that each of these is a vector space over the complex numbers. (Recall how complex numbers add and multiply: $(a_{0}+a_{1}i)+(b_{0}+b_{1}i)=(a_{0}+b_{0})+(a_{1}+b_{1})i$ and $(a_{0}+a_{1}i)(b_{0}+b_{1}i)=(a_{0}b_{0}-a_{1}b_{1})+(a_{0}b_{1}+a_{1}b_{0})i$.)

• The set of degree two polynomials with complex coefficients

• This set

  $\{\hskip 1.00006pt\begin{pmatrix}0&a\\ b&0\end{pmatrix}\mid a,b\in\mathbb{C}\text{\ and\ }a+b=0+0i\hskip 1.00006pt\}$

Name a property shared by all of the $\mathbb{R}^{n}$’s but not listed as a requirement for a vector space.

• Prove that for any four vectors $\vec{v}_{1},\ldots,\vec{v}_{4}\in V$ we can associate their sum in any way without changing the result.

  This allows us to write ‘$\vec{v}_{1}+\vec{v}_{2}+\vec{v}_{3}+\vec{v}_{4}$’ without ambiguity.

• Prove that any two ways of associating a sum of any number of vectors give the same sum. (Hint. Use induction on the number of vectors.)

Example 1.5 gives a subset of $\mathbb{R}^{2}$ that is not a vector space, under the obvious operations, because while it is closed under addition, it is not closed under scalar multiplication. Consider the set of vectors in the plane whose components have the same sign or are $0$. Show that this set is closed under scalar multiplication but not addition.