To verify whether each given function satisfies the criteria of an inner product, we need to check the following properties of an inner product on $\mathbb{R}^2$:

1. Linearity in the first argument: $\langle ax + by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$, where $a, b \in \mathbb{R}$ and $x, y, z \in \mathbb{R}^2$.
2. Symmetry: $\langle x, y \rangle = \langle y, x \rangle$.
3. Positivity: $\langle x, x \rangle \geq 0$ for all $x \in \mathbb{R}^2$, with equality if and only if $x = 0$.

Let’s analyze each function:

(a) $\langle x, y \rangle = x_1y_1 + x_2y_2$

1. Linearity: Expanding $\langle ax + by, z \rangle$: $\langle ax + by, z \rangle = \langle (ax_1 + by_1, ax_2 + by_2), z \rangle = a(x_1z_1 + x_2z_2) + b(y_1z_1 + y_2z_2),$ which equals $a\langle x, z \rangle + b\langle y, z \rangle$. Linearity holds.

2. Symmetry: $\langle x, y \rangle = x_1y_1 + x_2y_2 = y_1x_1 + y_2x_2 = \langle y, x \rangle.$ Symmetry holds.

3. Positivity: $\langle x, x \rangle = x_1^2 + x_2^2 \geq 0,$ with equality only if $x_1 = x_2 = 0$, i.e., $x = 0$. Positivity holds.

This is a valid inner product.

(b) $\langle v, v \rangle = v_1^2 + 3v_2^2$

1. Linearity: Expanding $\langle av + bw, z \rangle$: $\langle av + bw, z \rangle = a(v_1z_1 + 3v_2z_2) + b(w_1z_1 + 3w_2z_2),$ which equals $a\langle v, z \rangle + b\langle w, z \rangle$. Linearity holds.

2. Symmetry: $\langle v, w \rangle = v_1w_1 + 3v_2w_2 = w_1v_1 + 3w_2v_2 = \langle w, v \rangle.$ Symmetry holds.

3. Positivity: $\langle v, v \rangle = v_1^2 + 3v_2^2 \geq 0,$ with equality only if $v_1 = v_2 = 0$, i.e., $v = 0$. Positivity holds.

This is a valid inner product.

(c) $\langle S, t \rangle = 4S_1t_1 + 4S_2t_2 - S_1t_2 - S_2t_1$

1. Linearity: Expanding $\langle aS + bT, t \rangle$: $\langle aS + bT, t \rangle = 4(aS_1 + bT_1)t_1 + 4(aS_2 + bT_2)t_2 - (aS_1 + bT_1)t_2 - (aS_2 + bT_2)t_1,$ which simplifies to $a\langle S, t \rangle + b\langle T, t \rangle$. Linearity holds.

2. Symmetry: $\langle S, t \rangle = 4S_1t_1 + 4S_2t_2 - S_1t_2 - S_2t_1,$ but swapping $S$ and $t$: $\langle t, S \rangle = 4t_1S_1 + 4t_2S_2 - t_1S_2 - t_2S_1.$ These are not equal unless $S_1t_2 = S_2t_1$. Symmetry fails.

3. Positivity: For $S = t$, $\langle S, S \rangle = 4S_1^2 + 4S_2^2 - S_1S_2 - S_2S_1$, which may not always be $\geq 0$ (e.g., when $S_1$ or $S_2$ are negative). Positivity fails.

This is not a valid inner product.

Conclusion:

• (a) and (b) are valid inner products.
• (c) is not a valid inner product.

Would you like a detailed explanation of any specific step or concept?

Here are 5 related questions for practice:

1. Can you provide an example of a non-standard valid inner product on $\mathbb{R}^2$?
2. Why does symmetry fail for case (c)?
3. How does the Cauchy-Schwarz inequality relate to inner products?
4. Explain why the positivity condition ensures $\langle x, x \rangle$ represents a "magnitude."
5. How can inner products be generalized to complex vector spaces?

Tip: Always verify symmetry and positivity carefully; these are common points of failure in inner product verification.