Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, select all of the axioms that fail to hold. (Let $$ \mathbf{u}, \mathbf{v} $$, and $$ \mathbf{w} $$ be vectors in the vector space $$ V $$, and let $$ c $$ and $$ d $$ be scalars.) $$ \mathbb{R}^{2} $$, with the usual addition but scalar multiplication defined by $$ c\left[\begin{array}{l}x \ y\end{array}\right]=\left[\begin{array}{r}x \ c y\end{array}\right] \text {. } $$ All of the axioms hold, so the given set is a vector space. 1. $$ \mathbf{u}+\mathbf{v} $$ is in $$ V $$. 2. $$ \mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u} $$ 3. $$ (\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w}) $$ 4. There exists an element $$ \mathbf{0} $$ in $$ V $$, called a zero vector, such that $$ \mathbf{u}+\mathbf{0}=\mathbf{u} $$. 5. For each $$ \mathbf{u} $$ in $$ V $$, there is an element -u in $$ V $$ such that $$ \mathbf{u}+(-\mathbf{u})=\mathbf{0} $$. 6. $$ c \mathbf{u} $$ is in $$ V $$. 7. $$ c(\mathbf{u}+\mathbf{v})=c \mathbf{u}+c \mathbf{v} $$ 8. $$ (c+d) \mathbf{u}=c \mathbf{u}+d \mathbf{u} $$ 9. $$ c(d \mathbf{u})=(c d) \mathbf{u} $$

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To determine whether the given set $$ \mathbb{R}^{2} $$ with the specified operations is a vector space, we need to check if it satisfies all the vector space axioms. The operations defined are the usual addition and a modified scalar multiplication.

### Given Operations:
1. **Addition**: Usual vector addition in $$ \mathbb{R}^{2} $$.
2. **Scalar Multiplication**: Defined as 
   $$
   c\left[\begin{array}{l}x \ y\end{array}\right]=\left[\begin{array}{r}x \ c y\end{array}\right]
   $$

### Axioms to Check:
1. **Closure under addition**: For any $$ \mathbf{u}, \mathbf{v} \in V $$, $$ \mathbf{u} + \mathbf{v} $$ is in $$ V $$.
   - This holds true since the sum of two vectors in $$ \mathbb{R}^{2} $$ is also in $$ \mathbb{R}^{2} $$.

2. **Commutativity of addition**: $$ \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} $$.
   - This holds true for usual vector addition.

3. **Associativity of addition**: $$ (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) $$.
   - This holds true for usual vector addition.

4. **Existence of zero vector**: There exists an element $$ \mathbf{0} $$ in $$ V $$ such that $$ \mathbf{u} + \mathbf{0} = \mathbf{u} $$.
   - The zero vector in $$ \mathbb{R}^{2} $$ is $$ \left[\begin{array}{r}0 \ 0\end{array}\right] $$, and it satisfies this property.

5. **Existence of additive inverses**: For each $$ \mathbf{u} \in V $$, there exists an element $$ -\mathbf{u} \in V $$ such that $$ \mathbf{u} + (-\mathbf{u}) = \mathbf{0} $$.
   - This holds true since for any $$ \mathbf{u} = \left[\begin{array}{r}x \ y\end{array}\right] $$, we can take $$ -\mathbf{u} = \left[\begin{array}{r}-x \ -y\end{array}\right] $$.

6. **Closure under scalar multiplication**: $$ c \mathbf{u} $$ is in $$ V $$.
   - This holds true since $$ c \left[\begin{array}{l}x \ y\end{array}\right] = \left[\begin{array}{r}x \ c y\end{array}\right] $$ is still in $$ \mathbb{R}^{2} $$.

7. **Distributive property**: $$ c(\mathbf{u} + \mathbf{v}) = c \mathbf{u} + c \mathbf{v} $$.
   - Let's check this:
     $$
     c(\mathbf{u} + \mathbf{v}) = c\left(\left[\begin{array}{l}u_1 \ u_2\end{array}\right] + \left[\begin{array}{l}v_1 \ v_2\end{array}\right]\right) = c\left[\begin{array}{l}u_1 + v_1 \ u_2 + v_2\end{array}\right] = \left[\begin{array}{r}u_1 + v_1 \ c(u_2 + v_2)\end{array}\right]
     $$
     On the other hand,
     $$
     c \mathbf{u} + c \mathbf{v} = \left[\begin{array}{r}u_1 \ c u_2\end{array}\right] + \left[\begin{array}{r}v_1 \ c v_2\end{array}\right] = \left[\begin{array}{r}u_1 + v_1 \ c u_2 + c v_2\end{array}\right]
     $$
     These two expressions are not equal unless $$ c = 1 $$. Thus, this axiom fails.

8. **Distributive property of scalar addition**: $$ (c + d) \mathbf{u} = c \mathbf{u} + d \mathbf{u} $$.
   - Let's check this:
     $$
     (c + d) \mathbf{u} = \left[\begin{array}{r}u_1 \ (c + d) u_2\end{array}\right]
     $$
     On the other hand,
     $$
     c \mathbf{u} + d \mathbf{u} = \left[\begin{array}{r}u_1 \ c u_2\end{array}\right] + \left[\begin{array}{r}u_1 \ d u_2\end{array}\right] = \left[\begin{array}{r}u_1 + u_1 \ (c + d) u_2\end{array}\right]
     $$
     These two expressions are not equal. Thus, this axiom fails.

9. **Associativity of scalar multiplication**: $$ c(d \mathbf{u}) = (cd) \mathbf{u} $$.
   - Let's check this:
     $$
     d \mathbf{u} = \left[\begin{array}{r}u_1 \ d u_2\end{array}\right]
     $$
     Then,
     $$
     c(d \mathbf{u}) = c\left[\begin{array}{r}u_1 \ d u_2\end{array}\right] = \left[\begin{array}{r}u_1 \ c(d u_2)\end{array}\right] = \left[\begin{array}{r}u_1 \ (cd) u_2\end{array}\right]
     $$
     This holds true.

### Conclusion:
The axioms that fail to hold are:
- Axiom 7: $$ c(\mathbf{u} + \mathbf{v}) = c \mathbf{u} + c \mathbf{v} $$
- Axiom 8: $$ (c + d) \mathbf{u} = c \mathbf{u} + d \mathbf{u} $$

Thus, the given set $$ \mathbb{R}^{2} $$ with the specified operations is **not** a vector space.
The given set $$ \mathbb{R}^{2} $$ with the specified operations is **not** a vector space because axioms 7 and 8 fail to hold.

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