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 \( c\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{r}x \\ c y\end{array}\right] \). 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. \( c(d \mathbf{u})=(c d) \mathbf{u} \) 10. \( 1 \mathbf{u}=\mathbf{u} \)

We are given the set \( \mathbb{R}^2 \) with the usual addition \[ \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x+x' \\ y+y' \end{bmatrix}, \] but with scalar multiplication defined by \[ c\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ c y \end{bmatrix}. \] We will check the vector space axioms: 1. **Closure under addition:** For any two vectors \(\begin{bmatrix} x \\ y \end{bmatrix}\) and \(\begin{bmatrix} x' \\ y' \end{bmatrix}\), their sum is \[ \begin{bmatrix} x+x' \\ y+y' \end{bmatrix} \in \mathbb{R}^2. \] So this axiom holds. 2. **Commutativity of addition:** Since addition is the usual one, \[ \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x+x' \\ y+y' \end{bmatrix} = \begin{bmatrix} x'+x \\ y'+y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix} + \begin{bmatrix} x \\ y \end{bmatrix}. \] This axiom holds. 3. **Associativity of addition:** For any three vectors, the usual addition is associative. Thus, \[ (\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w}). \] This axiom holds. 4. **Existence of a zero vector:** The zero vector in \(\mathbb{R}^2\) is \(\begin{bmatrix} 0 \\ 0 \end{bmatrix}\) because \[ \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}. \] This axiom holds. 5. **Existence of additive inverses:** For any \(\begin{bmatrix} x \\ y \end{bmatrix}\), its additive inverse is \(\begin{bmatrix} -x \\ -y \end{bmatrix}\) since \[ \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} -x \\ -y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. \] This axiom holds. 6. **Closure under scalar multiplication:** For any scalar \( c \) and vector \( \begin{bmatrix} x \\ y \end{bmatrix} \), we have \[ c\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ c\,y \end{bmatrix}, \] which is clearly an element of \(\mathbb{R}^2\). This axiom holds. 7. **Distributivity of scalar multiplication with respect to vector addition:** Check that \[ c(\mathbf{u}+\mathbf{v}) = c\begin{bmatrix} x+x' \\ y+y' \end{bmatrix} = \begin{bmatrix} x+x' \\ c(y+y') \end{bmatrix} = \begin{bmatrix} x+x' \\ cy + cy' \end{bmatrix}, \] and \[ c\mathbf{u}+ c\mathbf{v} = \begin{bmatrix} x \\ cy \end{bmatrix} + \begin{bmatrix} x' \\ cy' \end{bmatrix} = \begin{bmatrix} x+x' \\ cy + cy' \end{bmatrix}. \] They are equal, so this axiom holds. 8. **Distributivity of scalar multiplication with respect to field addition:** We need to check whether \[ (c+d)\begin{bmatrix} x \\ y \end{bmatrix} = c\begin{bmatrix} x \\ y \end{bmatrix} + d\begin{bmatrix} x \\ y \end{bmatrix}. \] Compute the left-hand side: \[ (c+d)\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ (c+d)y \end{bmatrix}. \] Now compute the right-hand side: \[ c\begin{bmatrix} x \\ y \end{bmatrix} + d\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ cy \end{bmatrix} + \begin{bmatrix} x \\ dy \end{bmatrix} = \begin{bmatrix} x+x \\ cy+dy \end{bmatrix} = \begin{bmatrix} 2x \\ cy+dy \end{bmatrix}. \] For these to be equal, we require that \[ \begin{bmatrix} x \\ (c+d)y \end{bmatrix} = \begin{bmatrix} 2x \\ (c+d)y \end{bmatrix}. \] This forces \( x = 2x \), which holds only if \( x = 0 \). Since \( x \) can be any real number, this axiom fails in general. 9. **Compatibility of scalar multiplication with field multiplication (Associativity of scalar multiplication):** Check that \[ c\left(d\begin{bmatrix} x \\ y \end{bmatrix}\right) = c\begin{bmatrix} x \\ dy \end{bmatrix} = \begin{bmatrix} x \\ c(dy) \end{bmatrix} = \begin{bmatrix} x \\ (cd)y \end{bmatrix}, \] which is the same as \[ (cd)\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ (cd)y \end{bmatrix}. \] This axiom holds. 10. **Identity element of scalar multiplication:** Check that \[ 1\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ 1 \cdot y \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}. \] This axiom holds. The only axiom that fails is axiom 8, the distributivity of scalar multiplication with respect to field addition. Therefore, the given set with the specified operations is not a vector space, and axiom 8 fails.