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Problem 40: All upper triangular matrices of order n over $\R$. Vector space or not?

Problem 40: Is the following set form Vector space over reals?

All upper triangular matrices of order n over $\R$

Solution:
I. Properties under addition
i. Closure Property
Let $A,B \in \V$ = the vector spacr of upper traingular matrices.
i.e $\matAn{A}{a}$ and $\matAn{B}{b}$ then $a_{ij}, b_{ij} \in \R$ and $a_{ij} =0, b_{ij}=0$ for $i>j$.
Now $A+B = \matABn$, where $a_{ij}+b_{ij} =0+0=0$ for $i>j$
$\Rightarrow A+B$ is upper traingular matrix over reals.
$\therefore A+B \in \V$
$\Rightarrow \V$ is closed under addition.

ii. Associativity:
Let $A,B,C \in \V$ = the vector spacr of upper traingular matrices.
i.e $\matAn{A}{a}$,$\matAn{B}{b}$ and $\matAn{C}{c}$ then $a_{ij}, b_{ij},c_{ij} \in \R$ and $a_{ij} =0, b_{ij}=0, c_{ij}=0,$ for $i>j$.
\begin{align*} \left[A+B\right]+C = \left[\matn{a}+\matn{b}\right]+\matn{c}\\ a_{ij} =0, b_{ij}=0, c_{ij}=0,\;for\;i>j. =\matn{a}+\left[\matn{b}+\matn{c}\right]\\ =A+\left[B+C\right]\\ \end{align*} $\Rightarrow$ Addition is associative.

iii. Existence of additive identity:
Let \begin{align*} O=\left[0\right]_{n \times n} \end{align*} Now \begin{align*} A+O=\matn{a}+\left[0\right]_{n \times n}\\ =\matnij{a_{ij}+0}\\ =\matn{a}\\ = A\\ \end{align*} and \begin{align*} O+A=\left[0\right]_{n \times n}+\matn{a}\\ =\matnij{0+a_{ij}}\\ =\matn{a}\\ = A\\ \therefore A+O=A=O+A, a_{ij} =0, b_{ij}=0, c_{ij}=0,\;for\;i>j\\ \end{align*} $\Rightarrow O$ is the additive identity.

iv. Existence of additive inverse:
let \begin{align*} \matAn{A}{a} \in \V\\ \matAn{-A}{-a} \in \V \end{align*} Now \begin{align*} A+(-A)==\matn{a}+\matn{-a}\\ =\matnij{a_{ij}+(-a_{ij})}\\ =O\\ \end{align*} and \begin{align*} (-A)+A=\matn{-a}+\matn{a}\\ =\matnij{(-a_{ij})+a_{ij}}\\ =O\\ \therefore A+(-A)=O=(-A)+A a_{ij} =0\;for\;i>j\\ \end{align*} $\Rightarrow -A$ is the additive inverse.

v. Commutative Law:
Let \begin{align*} \matAn{A}{a} \in \V\\ \matAn{B}{b} \in \V\\ \end{align*} Now \begin{align*} A+B=\matn{a}+\matn{b}\\ =\matABn\\ =\matBAn\\ =\matn{b}+\matn{a}\\ = B+A\\ \end{align*} $\Rightarrow$ addition is commutative.

II. Properties under Scalar Multiplication
vi. Closure under Scalar Multiplication
Let $\alpha\in \R$ then $\alpha A = \alpha\matn{a}$. where $\alpha a_{ij} \in \R$ and \begin{align*} \alpha\matn{a} = \alpha.0 = 0 \forall i>j\\ \end{align*} $\therefore \alpha A$ is upper traingular matrix. $\Rightarrow \V$ is closed under Scalar Multiplication.

vii. Distributivity of Addition of Vectors
Let $\alpha\in \R$ and $ \matAn{A}{a}, \matAn{B}{b}\in \V\\$ \begin{align*} \alpha\left[A+B\right]= \alpha\left[\matn{a}+\matn{b}\right]\\ = \alpha\left[\matABn\right]\\ = \matnij{\alpha a_{ij}+\alpha b_{ij}}\\ = \matnij{\alpha a_{ij}}+\matnij{\alpha b_{ij}}\\ =\alpha \matn{a} + \alpha \matn{b}\\ =\alpha A+\alpha B\\ \end{align*} $\Rightarrow $ Distributivity of Addition of Vectors exists.

viii.Distributivity of Addition of Real Numbers
Let $\alpha, \beta\in \R$ and $\matAn{A}{a} \in \V$ \begin{align*} (\alpha+\beta)A= (\alpha+\beta)\matn{a}\\ = \matnij{(\alpha+\beta) a_{ij}}\\ = \matnij{\alpha a_{ij}+\beta a_{ij}}\\ = \matnij{\alpha a_{ij}}+\matnij{\beta a_{ij}}\\ =\alpha(\matn{a})+\beta(\matn{b})\\ =\alpha A+\beta B\\ \end{align*} $\Rightarrow $ Distributivity of Addition of Real Numbers exists.

ix.Associativity of multiplication
Let $\alpha, \beta\in \R$ and $\matAn{A}{a} \in \V$ \begin{align*} (\alpha\beta)A= (\alpha\beta)\matn{a}\\ = \matn{(\alpha\beta) a}\\ = \matn{\alpha(\beta a)}\\ = \alpha\matn{(\beta a)} = \alpha \left[\beta \matn{a}\right]\\ =a(bA) \end{align*} $\Rightarrow$ Associativity of multiplication exists.

x. Unity element multiplication
Let 1 be unity element of field $\F$ and $\matAn{A}{a}\in \V$
Now \begin{align*} 1.A= 1.\matn{a}\\ =\matn{1.a}\\ =\matn{a}\\ = A \end{align*} $\Rightarrow 1.A = A$.

Hence, $\V$ is a vector space over $\R$.

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