Question

Prove that \(\|\lambda A\|=|\lambda|\|A\|,\|A+B\| \leq\|A\|+\|B\|\), and \(\|A B\| \leqq\) \(\|A\|\|B\|\), where \(A: R^{n} \rightarrow R^{n}\) and \(B: R^{n} \rightarrow \boldsymbol{R}^{n}\) are linear operators, and \(\lambda \in \overline{\boldsymbol{R}}\). is a number.

Short answer

#Conclusion# In this exercise, we have proven the following three properties of norm for linear operators: 1. \(\|\lambda A\|=|\lambda|\|A\|\) - The norm of a scalar multiplied by a linear operator is equal to the absolute value of the scalar multiplied by the norm of the linear operator. 2. \(\|A+B\| \leq\|A\|+\|B\|\) - The norm of the sum of two linear operators is less than or equal to the sum of their norms (Triangle Inequality). 3. \(\|AB\| \leq \|A\|\|B\|\) - The norm of the product of two linear operators is less than or equal to the product of their norms. These properties help to describe and analyze the behavior of linear operators in various settings, such as functional analysis and linear algebra.

Step-by-step solution

Prove \(\|\lambda A\|=|\lambda|\|A\|\)

Suppose x is an arbitrary vector in R^n. We need to prove that the norm of the linear operator λA acting on x is equal to |λ| times the norm of the linear operator A acting on x. In other words, we need to show that: \(\|\lambda A\| = |\lambda| \|A\|\) Using the definition of the norm, we can write the left side as: \(\|\lambda A\| = \sup_{x\ne0}\frac{\|\lambda Ax\|}{\|x\|}\) Now, we can use the fact that λAx = λ(Ax), which implies: \(\|\lambda A\| = \sup_{x\ne0}\frac{\|\lambda(Ax)\|}{\|x\|} = \sup_{x\ne0}\frac{|\lambda|\|Ax\|}{\|x\|}\) Since |λ| is constant, we can move it outside of the supremum: \(\|\lambda A\| = |\lambda|\sup_{x\ne0}\frac{\|Ax\|}{\|x\|}\) But, the expression inside the supremum is the definition of the norm of A: \(\|\lambda A\| = |\lambda|\|A\|\) This proves the first property.

Prove \(\|A+B\| \leq\|A\|+\|B\|\)

We need to prove the triangle inequality for linear operators, which states that: \(\|A+B\| \leq \|A\|+\|B\|\) Let x be an arbitrary vector in R^n. Consider the inequality: \(\|A(x) + B(x)\| \leq \|Ax\| + \|Bx\|\) This inequality holds due to the triangle inequality property of vector norms. Now, divide both sides by \(\|x\|\) (assuming x ≠ 0): \(\frac{\|(A + B)x\|}{\|x\|} \leq \frac{\|Ax\|}{\|x\|} + \frac{\|Bx\|}{\|x\|}\) We can now use the definition of norms: \(\|A+B\| \leq \|A\| + \|B\|\) This proves the second property.

Prove \(\|AB\| \leqq \|A\|\|B\|\)

We need to prove that the norm of the product of two linear operators is less than or equal to the product of their norms: \(\|AB\| \leqq \|A\|\|B\|\) Let x be an arbitrary vector in R^n and y = Bx. Then, we can write the inequality: \(\|Ay\| \leq \|A\|\|y\|\) Divide both sides by \(\|y\|\) (assuming y ≠ 0): \(\frac{\|Ay\|}{\|y\|} \leq \|A\|\) Now, as y = Bx, we substitute it back to the inequality: \(\frac{\|A(Bx)\|}{\|Bx\|} \leq \|A\|\) Multiply both sides by \(\frac{\|Bx\|}{\|x\|}\) (assuming x ≠ 0): \(\frac{\|ABx\|}{\|x\|} \leq \|A\|\frac{\|Bx\|}{\|x\|} \leq \|A\|\|B\|\) Using the definition of norms, we have: \(\|AB\| \leqq \|A\|\|B\|\) This proves the third property.

Key concepts

Linear Operators

Linear operators are fundamental in linear algebra, acting as functions that map one vector from a vector space to another, while preserving vector addition and scalar multiplication. Simply put, if you have a linear operator \( A \), and vectors \( u \) and \( v \), the operator satisfies two main properties:

• Additivity: \( A(u + v) = Au + Av \).
• Homogeneity: \( A(cu) = c(Au) \), where \( c \) is a scalar.

These properties make linear operators predictable and easy to manage within vector spaces, as they ensure that the operations adhere to linear rules. Understanding this concept is crucial as it lays the groundwork for exploring more complex operations involving vectors.

Vector Norms

Vector norms provide a way to quantify the "size" or "length" of a vector, and are a critical tool in many linear algebra applications. A vector norm is typically denoted as \( \|x\| \) and satisfies three conditions:

• Non-negativity: \( \|x\| \geq 0 \). A norm is always non-negative, and \( \|x\| = 0 \) only if \( x \) is the zero vector.
• Scalar multiplication: \( \|c x\| = |c| \|x\| \), where \( c \) is a scalar. This indicates that scaling a vector by a factor \( c \) scales its norm by the absolute value of \( c \).
• Triangle inequality: \( \|x + y\| \leq \|x\| + \|y\| \). This property is shared with the triangle inequality in geometry, illustrating the intuitive point that the direct path between two points (via \( x + y \)) is the shortest.

Different norms can be defined, such as the 1-norm, 2-norm (Euclidean norm), and infinity norm, each providing unique perspectives on vector measurement.

Triangle Inequality

The triangle inequality is a principle that asserts the sum of the "lengths" of two sides of a triangle is greater than or equal to the "length" of the third side. For vectors, this is expressed as \( \|x + y\| \leq \|x\| + \|y\| \), emphasizing that taking two vectors and adding them will not exceed the sum of their individual norms.

Understanding this inequality is crucial in studying vector spaces as it provides a foundation for examining distances and establishing stability and error bounds in numerical methods. It's a cornerstone in the proofs related to norms of operators, ensuring we understand how these intrinsic properties manifest in both theoretical and practical applications.

Inequalities in Norms

Inequalities in norms refer to various inequalities that vector norms must satisfy, ensuring mathematical consistency and reliability when manipulating vectors and operators. These inequalities include:

• Scalar multiplication norm equality: For a scalar \( \lambda \) and a vector \( x \), we have \( \|\lambda x\| = |\lambda| \|x\| \). This confirms that scaling a vector by a scalar \( \lambda \) proportionally scales its norm by the absolute value of \( \lambda \).
• Operator norm inequalities: For linear operators \( A \) and \( B \), \( \|A + B\| \leq \|A\| + \|B\| \) reflects the triangle inequality, while \( \|AB\| \leq \|A\|\|B\| \) ensures the combined effect of operators does not exceed their individual impacts.

These inequalities are pivotal in linear algebra and functional analysis, as they govern how linear systems behave and how precise conclusions can be derived when analyzing these systems.