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(See Solution) In each of the following, determine whether or not is an inner product in the given vector space: For u and v in R^2 put =2 u_1

Question: In each of the following, determine whether or not $\langle-,-\rangle$ is an inner product in the given vector space:

For $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{2}$ put $\langle\mathbf{u}, \mathbf{v}\rangle=2 u_{1} v_{1}-u_{2} v_{2}$;
For $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{2}$ put $\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{1} v_{2}+u_{2} v_{2} ;$
For $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{2}$ put $\langle\mathbf{u}, \mathbf{v}\rangle=u_{1}^{2} v_{1}^{2}+u_{2}^{2} v_{2}^{2}$.

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(See Solution) In each of the following, determine whether or not is an inner product in the given vector space: For u and v in R^2 put =2 u_1

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