Question

determine whether the following is an inner product:
(x,y)=\[2x _{1}y _{1}+x _{1}y _{2}+3x _{2}y _{2}-x _{2} y _{1}\]

Answer 1

i'll have to read up on this - i dont know waht you mean by an inner product

Answer 2

Inner product?
Sorry but truthfully i heard it for the first time

Answer 3

me too

Answer 4

You need to verify that it satisfies the inner product axioms:

1) Conjugate symmetry: \(\langle x, y\rangle = \overline{\langle y, x\rangle} \)
2) Linearity in the first argument: \(\langle ax + y, z\rangle = a \langle x, z \rangle + \langle y, z \rangle\)
3) Positive-definiteness: \(\langle x, x \rangle \geq 0\) and 0 only when \(x = 0\)

Answer 5

look at <(1,0),(0,1)> and <(0,1),(1,0)>

Answer 6

is it enough to look at one example? ive done what alchemists said but am not sure if i have come to the correct conclusion? is it an inner product?

Answer 7

\[\langle (1, 0), (0, 1) \rangle = 2(1)(0) + (1)(1) + 3(0)(1) - (0)(0) = 1\]\[\langle (0, 1), (1, 0) \rangle = 2(0)(1) + (0)(0) + 3(1)(0) - (1)(1) = -1\]So \(\langle x, y \rangle = \langle (1, 0), (0, 1) \neq \langle (0, 1), (1, 0) \rangle = \overline{\langle y, x \rangle}\), therefore it is not an inner product.