Producto interno - Ejercicio resuelto - ¿Es un producto interno de R2 la función (v,w)=8v1w1-3v2w2?

Summary

The video is abnormal, and we are working hard to fix it.
Please replace the link and try again.

Takeaways

• 😀 The exercise involves determining whether a given function defines an inner product on the vector space R².
• 😀 The function in question is: f(v, w) = 8v₁w₁ - 3v₂w₂, where v₁, v₂ and w₁, w₂ are the components of vectors v and w, respectively.
• 😀 To verify if the function is a valid inner product, it must be checked against the key properties of inner products.
• 😀 The first property of an inner product states that the inner product of a vector with itself should be greater than or equal to zero.
• 😀 A common misconception about inner products is that they must always yield non-negative results; however, this is not true for all pairs of vectors, only when a vector is paired with itself.
• 😀 For the given function, when testing it with specific vectors, the result of f(v, v) is not always non-negative.
• 😀 A counterexample is provided where vector v = (0, 1) results in f(v, v) = -3, which violates the first property of an inner product.
• 😀 The result of f(v, v) for v = (0, 1) demonstrates that the function can yield negative results, disproving that the function satisfies the first property.
• 😀 The violation of the first property means that the function does not define a valid inner product.
• 😀 The exercise concludes that the function fails to meet the necessary conditions to be an inner product because it does not satisfy the required properties, specifically the non-negativity of the inner product of a vector with itself.

Q & A

What is the primary objective of the exercise discussed in the script?

The primary objective is to verify whether a given function defines an inner product on the space R2, by checking it against the properties of an inner product.

What does the function in the script take as inputs, and what does it return?

The function takes two vectors from R2 as inputs and returns a scalar as the output.

How is the first property of an inner product, which requires that the product of a vector with itself is greater than or equal to zero, tested in this script?

The script checks if the inner product of a vector with itself results in a non-negative value. It correctly explains that the square of real numbers will always be non-negative, but also notes that a negative factor could result in a negative outcome in some cases.

What clarification does the speaker provide regarding the common misconception about inner products?

The speaker clarifies that the inner product of two different vectors can indeed be negative, and the common misconception that the inner product is always non-negative only applies when a vector is multiplied by itself.

How does the speaker demonstrate the failure of the first property of the inner product in this case?

The speaker demonstrates the failure by showing that depending on the values of the vectors, the result of the inner product can be negative, especially if the negative term outweighs the positive one.

What is the example chosen by the speaker to show that the first property does not hold?

The speaker chooses the vector (0,1) in R2, where the positive term of the inner product is zero, and the negative term results in a negative value when computed, thus violating the property that the product of a vector with itself should be non-negative.

What specific values for the vector components are chosen in the example to show the failure of the property?

The speaker chooses the vector (0,1), which results in a positive term of zero and a negative term of -3, leading to a negative inner product value.

What mathematical operation does the speaker use to compute the inner product in the example?

The speaker computes the inner product using the formula: 8 * v1^2 - 3 * v2^2, where v1 and v2 represent the components of the vector.

What is the final outcome of the example calculation?

The final outcome of the example calculation is -3, which shows that the inner product of the vector with itself is negative, thus proving that the first property does not hold.

What conclusion does the speaker draw from this example?

The speaker concludes that the given function does not define a valid inner product because it fails to satisfy the first property, which requires that the inner product of a vector with itself is always non-negative.