Given:
Vector space = \( \mathbb{R}^2(\mathbb{R}) \)
Formula used:
An inner product \( \langle u, v \rangle \)` must satisfy:
- Symmetry: \( \langle u, v \rangle = \langle v, u \rangle \)`
- Linearity: \( \langle au + bv, w \rangle = a\langle u, w \rangle + b\langle v, w \rangle \)`
- Positive Definiteness: \( \langle u, u \rangle > 0 \)
for all \\( u \neq 0 \\)
Calculation:
Option (1): \( \langle (x_1, y_1), (x_2, y_2) \rangle = x_1x_2 + x_1y_2 + 4x_2y_1 + y_1y_2 \)`
⇒ \( \langle (1, 0), (0, 1) \rangle = 1 \times 0 + 1 \times 1 + 4 \times 0 \times 0 + 0 \times 1 = 1 \)`
⇒ \( \langle (0, 1), (1, 0) \rangle = 0 \times 1 + 0 \times 0 + 4 \times 1 \times 1 + 1 \times 0 = 4 \)`
⇒ \( 1 \neq 4 \)` (Not symmetric)
Option (2): Contains \( x_1^2x_2^2 \) and \\( y\_1^2y\_2^2 \\)
⇒ Violates linearity (powers of 2)
Option (3): \( \langle (x, y), (x, y) \rangle = 4x^2 + xy + xy = 4x^2 + 2xy \)`
⇒ For \( (x, y) = (0, 1) \), \\( \langle u, u \rangle = 4(0)^2 + 2(0)(1) = 0 \\)
⇒ Fails positive definiteness
Option (4): \( \langle (x_1, y_1), (x_2, y_2) \rangle = x_1x_2 + x_1y_2 + x_2y_1 + y_1y_2 \)`
⇒ \( \langle (x, y), (x, y) \rangle = x^2 + 2xy + y^2 = (x + y)^2 \)`
⇒ Option (4) is a symmetric bilinear form.
∴ The correct answer is option (4).