Question

Under what conditions does equality hold in the Schwarz inequality?

Short answer

Answer: Equality holds in the Cauchy-Schwarz (Schwarz) Inequality if and only if the vectors u and v are linearly dependent, meaning one vector is a scalar multiple of the other (i.e., u = k*v for some constant k).

Step-by-step solution

State the Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality is given by: \(| \textbf{u} \cdot \textbf{v} | \leq ||\textbf{u}|| ||\textbf{v}||\), where \(\textbf{u}\) and \(\textbf{v}\) are two arbitrary vectors and \(||\textbf{u}||\) and \(||\textbf{v}||\) denotes their magnitudes. Here, the dot product \(\textbf{u}\cdot\textbf{v}\) is equivalent to the sum of the product of their respective components: \(\textbf{u}\cdot\textbf{v} = u_1v_1+u_2v_2+\cdots +u_nv_n\)

Find when equality holds

Let's consider when \(\textbf{u} = k \textbf{v}\) for some constant \(k\). In this case, the dot product becomes \(k(v_1^2+v_2^2+\cdots +v_n^2)\) and the magnitudes become \(|k| \sqrt{v_1^2+v_2^2+\cdots +v_n^2}\) and \(\sqrt{v_1^2+v_2^2+\cdots +v_n^2}\). The inequality now becomes: \(|k(v_1^2+v_2^2+\cdots +v_n^2)| \leq |k| (v_1^2+v_2^2+\cdots +v_n^2)\) Since the left side and the right side of the inequality are the same, the equality holds when \(\textbf{u} = k \textbf{v}\). In other words, equality holds in the Schwarz inequality when the two vectors are linearly dependent.

Conclusion

The equality in the Cauchy-Schwarz (Schwarz) Inequality holds if and only if the vectors \(\textbf{u}\) and \(\textbf{v}\) are linearly dependent. This means that one vector is a scalar multiple of the other, i.e., \(\textbf{u} = k \textbf{v}\) for some constant \(k\).

Key concepts

Vectors

Vectors are fundamental in understanding many concepts in mathematics and physics. A vector is a quantity that has both a magnitude and a direction. Think of it as an arrow that points in a certain direction and has a certain length.

When we talk about vectors in the context of the Cauchy-Schwarz Inequality, we consider their properties like

• Magnitude: This is the length of the vector, often represented by \(||\textbf{v}||\).
• Direction: The arrow direction signifies the line along which the vector acts.
• Components: These describe the vector in a coordinate system, such as \(u_1, u_2, ... , u_n\) for vector \(\textbf{u}\).

In operations, vectors can be added, subtracted, or multiplied by scalars. For example, in the Cauchy-Schwarz Inequality, the dot product operation is used, which combines vectors in a special way. This involves multiplying corresponding components of the vectors and summing them up to give a single number. Understanding the basic concepts of vectors is crucial to grasping how inequalities like Cauchy-Schwarz work.

Linear Dependence

Linear dependence is a crucial concept in linear algebra, especially when understanding when equality holds in the Cauchy-Schwarz Inequality. Two vectors are linearly dependent if one is a scalar multiple of the other. In other words,

• One vector can be expressed as \(\textbf{u} = k \textbf{v}\), where \(k\) is a constant.
• This relationship implies that both vectors point in the same or exact opposite direction.

Linearly dependent vectors don't offer new "directions" in space. If you know one vector, you can find others in the set through multiplication by scalars. Importantly, in the case of the Cauchy-Schwarz Inequality, the equality only holds true if the vectors \(\textbf{u}\) and \(\textbf{v}\) are linearly dependent. This relationship simplifies the inequality, turning it into an actual equation when the two vectors reach the state of maximum equality.

Scalar Multiples

Scalar multiples play a significant role in determining vector relationships under the Cauchy-Schwarz Inequality. A scalar is a real number that "stretches" or "shrinks" a vector. Making one vector a scalar multiple of another involves scaling its components with a constant factor. This is represented mathematically as:

• \(\textbf{u} = k \textbf{v}\), where \(k\) is the scalar.

When vectors are scalar multiples of each other, they maintain their direction but vary in magnitude based on the scalar. In solving problems involving the Cauchy-Schwarz Inequality, identifying scalar multiples is key to recognizing when equality is satisfied. When one vector is a scalar multiple of the other, the dot product equals the product of their magnitudes, achieving the equality condition. This concept illustrates how scalar multiplication influences vector magnitude and plays a critical role in vector analysis.