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Problem 47: Prove that the set of all solutions to the homogeneous linear system $Ax=0$ is a vector space

Problem 47: Prove that the set of all solutions to the homogeneous linear system $Ax=0$ is a vector space

Solution: Given $\V = \{x\R^n:Ax =0,$ where $A$ is a fixed matrix $\}$

I. Properties under addition
i. Closure Property
Let $u,v \in \V$ \begin{align*} A(u+v) = Au+Av \\ = 0+0=0\\ \therefore u+v \in \V. \end{align*} $\Rightarrow \V$ is closed under addition.

ii. Associativity:
Let $u,v,w \in \V$ Now, \begin{align*} A(\left[u+v\right]+w) = A(\left[(u+v+w)\right])\\ = Au+Av+Aw\\ = 0+0+0 = 0\\ \end{align*} And, \begin{align*} A(u+\left[(v+w\right]) = A(\left[(u+v+w)\right])\\ = Au+Av+Aw\\ = 0+0+0 = 0\\ \therefore A(\left[u+v\right]+w) = A(u+\left[(v+w\right]) \end{align*} $\Rightarrow$ Addition is associative in $\V$.

iii. Existence of additive identity:
For all $u \in \V, \exists 0 \in \V$
Now \begin{align*} A(u+0)=Au+A0 = 0 +0 = 0\\ A(0+u)=A0+Au = 0 +0 = 0\\ \end{align*} $\Rightarrow 0$ is the additive identity in $\V$.

iv. Existence of additive inverse:
Let $u \in \V$, be any element $-u\in \V$
Now \begin{align*} A(u+-u)= Au+ A(-u)=Au - Au = 0-0=0\\ A(-u+u)= A(-u)+Au=-Au+Au = -0+0 = 0\\ \end{align*} $\Rightarrow -u$ is the additive inverse of $u$ for each $u \in \V$.

v. Commutative Law:
Let $ u,v \in \V$, Now \begin{align*} A(u+v)= Au+Av\\ = Av+Au = A(v+u) \end{align*} $\Rightarrow$ addition is commutative in $\V$.

II. Properties under Scalar Multiplication
vi. Closure under Scalar Multiplication
Let $\alpha\in \R$ and $u \in \V$ \begin{align*} A(\alpha u) = \alpha Au = \alpha 0 = 0\\ \therefore \alpha u \in \V \end{align*} $\Rightarrow \V$ is closed under Scalar Multiplication.

vii.Distributivity of Addition of Real Numbers
Let $\alpha, \beta\in \R$ and $u \in \V$ \begin{align*} A((\alpha+\beta)u)= (\alpha+\beta)Au =(\alpha+\beta)0 =0\\ A(\alpha u) +A(\beta u) = \alpha Au+\beta Au = \alpha 0 + \beta 0 =0+0=0 \end{align*} $\Rightarrow A((\alpha+\beta)u) = A(\alpha u)+A(\beta u)$.
$\Rightarrow $ Distributivity of Addition of Real Numbers does exists.

viii. Distributivity of Addition of Vectors
Let $\alpha\in \R$ and $u, v \in \V$ Now, \begin{align*} A(\alpha\left[u+v\right])= A(\alpha u + \alpha v)\\ = A(\alpha u) + A(\alpha v)\\ =\alpha Au+\alpha Av\\ = \alpha 0 + \alpha 0\\ =0+0 =0 \end{align*} And,
\begin{align*} A(\alpha u) +A(\alpha v) =\alpha Au+\alpha Av\\ = \alpha 0 + \alpha 0\\ = 0+0 =0 \end{align*} $\Rightarrow $ Distributivity of Addition of Vectors exists.

ix.Associativity of multiplication
Let $\alpha, \beta\in \R$ and $u\in \V$ \begin{align*} A((\alpha\beta)u)= (\alpha\beta)Au\\ = \alpha(\beta Au\\ =\alpha(\beta Au) \end{align*} $\Rightarrow$ Associativity of multiplication exists.

x. Unity element multiplication
Let 1 be unity element of field $\F$ and $u\in \V$
Now \begin{align*} A(1\cdot u)= A1\cdot Au\\ = A1\cdot 0 = 0 \end{align*} $\Rightarrow 1\cdot u = u \forall u \in \V$.

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

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