Consider the mapping $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by \[ T\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{c} x^{2} \\ x+y \end{array}\right) \] Is $T$ a linear map? [] Yes O No

Step 1 ：Consider the mapping \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) given by \(T\left(\begin{array}{l} x \ y \end{array}\right)=\left(\begin{array}{c} x^{2} \ x+y \end{array}\right)\). We need to determine if \(T\) is a linear map.

Step 2 ：A linear map, or linear transformation, satisfies two properties: 1. Additivity: \(T(u + v) = T(u) + T(v)\) for all vectors \(u\) and \(v\) in the vector space. 2. Scalar multiplication: \(T(cu) = cT(u)\) for all vectors \(u\) in the vector space and all scalars \(c\).

Step 3 ：Let's consider two arbitrary vectors in \(\mathbb{R}^{2}\), say \(u = (a, b)\) and \(v = (c, d)\), and a scalar \(k\). We can then apply the transformation \(T\) to these vectors and check if the properties of additivity and scalar multiplication hold.

Step 4 ：For additivity, we have \(T(u + v) = ((a + c)^{2}, a + b + c + d)\) and \(T(u) + T(v) = (a^{2} + c^{2}, a + b + c + d)\). We can see that the additivity property does not hold because \((a + c)^{2}\) is not equal to \(a^{2} + c^{2}\).

Step 5 ：For scalar multiplication, we have \(T(ku) = (a^{2}k^{2}, a*k + b*k)\) and \(kT(u) = (a^{2}*k, k*(a + b))\). We can see that the scalar multiplication property does not hold because \(a^{2}*k^{2}\) is not equal to \(a^{2}*k\).

Step 6 ：Therefore, the transformation \(T\) does not satisfy the properties of a linear map.

Step 7 ：\(\boxed{\text{Final Answer: No, } T \text{ is not a linear map.}}\)