Problem 37 - Advanced Engineering Mathematics

Question

If \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w}\) are vectors in a vector space \(V\), then the axioms of an inner product \((\mathbf{u}, \mathbf{v})\) are (i) \((\mathbf{u}, \mathbf{v})=(\mathbf{v}, \mathbf{u})\) (ii) \((k \mathbf{u}, \mathbf{v})=k(\mathbf{u}, \mathbf{v}), k\) a scalar (iii) \((\mathbf{u}, \mathbf{u})=0\) if \(\mathbf{u}=\mathbf{0}\) and \((\mathbf{u}, \mathbf{u})>0\) if \(\mathbf{u} \neq \mathbf{0}\) (iv) \((\mathbf{u}, \mathbf{v}+\mathbf{w})=(\mathbf{u}, \mathbf{v})+(\mathbf{u}, \mathbf{w})\) Show that \((\mathbf{u}, \mathbf{v})=u_{1} v_{1}+4 u_{2} v_{2}\), where \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle\) and \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), is an inner product on \(R^{2} .\)

Step-by-Step Solution

Answer

The given function is an inner product on \(\mathbb{R}^2\).

Step 1: Verify Symmetry Axiom

The symmetry axiom states that \((\mathbf{u}, \mathbf{v}) = (\mathbf{v}, \mathbf{u})\). For our given expression \((\mathbf{u}, \mathbf{v}) = u_1 v_1 + 4 u_2 v_2\), we must check if this equals \((\mathbf{v}, \mathbf{u})\). Compute: \((\mathbf{v}, \mathbf{u}) = v_1 u_1 + 4 v_2 u_2\). Both expressions are equal because the real numbers are commutative, thus confirming symmetry: \(u_1 v_1 + 4 u_2 v_2 = v_1 u_1 + 4 v_2 u_2\).

Step 2: Verify Linearity in First Argument Axiom

The linearity axiom in the first argument states \((k \mathbf{u}, \mathbf{v}) = k(\mathbf{u}, \mathbf{v})\) for a scalar \(k\). Compute \((k\mathbf{u}, \mathbf{v}) = (k u_1, k u_2) \cdot (v_1, v_2) = k u_1 v_1 + 4k u_2 v_2\). On the other hand, \(k(\mathbf{u}, \mathbf{v}) = k(u_1 v_1 + 4 u_2 v_2) = k u_1 v_1 + 4k u_2 v_2\). Both expressions match, thus confirming linearity in the first argument.

Step 3: Verify Positive Definiteness Axiom

The positive definiteness axiom states \((\mathbf{u}, \mathbf{u}) = 0\) if \(\mathbf{u} = \mathbf{0}\) and \((\mathbf{u}, \mathbf{u}) > 0\) if \(\mathbf{u} eq \mathbf{0}\). Consider \((\mathbf{u}, \mathbf{u}) = u_1^2 + 4 u_2^2\). If \(\mathbf{u} = \mathbf{0}\), then \(u_1 = u_2 = 0\) giving \(0 + 0 = 0\). If \(\mathbf{u} eq \mathbf{0}\), then at least one of \(u_1\) or \(u_2\) is non-zero. Hence, \(u_1^2 + 4u_2^2 > 0\), satisfying the positive definiteness condition.

Step 4: Verify Linearity in Second Argument (Additivity) Axiom

The additivity (linearity in its second argument) axiom states \((\mathbf{u}, \mathbf{v} + \mathbf{w}) = (\mathbf{u}, \mathbf{v}) + (\mathbf{u}, \mathbf{w})\). Let \(\mathbf{v} = \langle v_1, v_2 \rangle\) and \(\mathbf{w} = \langle w_1, w_2 \rangle\). Compute \(\mathbf{v} + \mathbf{w} = \langle v_1 + w_1, v_2 + w_2 \rangle\) leading to \((\mathbf{u}, \mathbf{v} + \mathbf{w}) = u_1(v_1 + w_1) + 4u_2(v_2 + w_2) = u_1 v_1 + 4u_2 v_2 + u_1 w_1 + 4u_2 w_2\). This equals \((\mathbf{u}, \mathbf{v}) + (\mathbf{u}, \mathbf{w}) = (u_1 v_1 + 4u_2 v_2) + (u_1 w_1 + 4u_2 w_2)\). Hence, additivity is verified.

Key Concepts

Axioms of Inner Products

In the context of linear algebra, understanding inner product spaces crucially revolves around certain fundamental axioms. An inner product is essentially a way to multiply two vectors, resulting in a scalar. For a function to qualify as an inner product, it must satisfy four key axioms: symmetry, linearity in the first argument, linearity in the second argument (additivity), and positive definiteness. These properties form the backbone of inner product spaces, allowing advanced operations such as length and angle calculations in vector spaces.
Let's think of an inner product like a sophisticated calculator for vectors. By applying the axioms, we transform abstract concepts into precise mathematical structures. Exploring these axioms provides the foundational understanding necessary to work with inner products effectively.

Positive Definiteness

The concept of positive definiteness ensures that the inner product of a vector with itself is always non-negative. In simpler terms, this means the result of \( (\mathbf{u}, \mathbf{u}) \) should always be zero if the vector is a zero vector, or greater than zero if it's a non-zero vector.

• If \( \mathbf{u} = \mathbf{0} \), where all components are zero, then \( (\mathbf{u}, \mathbf{u}) = 0 \).
• If \( \mathbf{u} eq \mathbf{0} \), then \( (\mathbf{u}, \mathbf{u}) > 0 \).

This axiom guarantees that vectors have a meaningful ‘length’ and that the concept of distance or magnitude is well-defined. This is pivotal in various applications, ensuring that mathematical operations make intuitive sense.

Linearity

Linearity in inner products splits into two parts, known as the linearity in the first argument and linearity in the second argument (additivity). It states that when you apply any scalar multiple to the first vector, that scalar can be factored out of the inner product calculation.

• For any scalar \( k \), the axiom of linearity in the first argument holds: \( (k \mathbf{u}, \mathbf{v}) = k(\mathbf{u}, \mathbf{v}) \).
• Additivity deals with sum operations in vectors, where \( (\mathbf{u}, \mathbf{v} + \mathbf{w}) = (\mathbf{u}, \mathbf{v}) + (\mathbf{u}, \mathbf{w}) \).

Linearity ensures that when vectors and scalars interact, the process remains consistent and predictable, maintaining structure and form across calculations.

Symmetry

Symmetry in the context of an inner product means that the order of the vectors in the operation does not matter; the inner product of \( \mathbf{u} \) and \( \mathbf{v} \) is the same as that of \( \mathbf{v} \) and \( \mathbf{u} \). In terms of equations, this means:\[(\mathbf{u}, \mathbf{v}) = (\mathbf{v}, \mathbf{u})\]Symmetry is a straightforward yet vital property because it ensures that the order of operations does not affect the outcome. This consistency is essential when dealing with complex systems, allowing us to exchange vectors in equations without altering the result. The commutative nature of symmetry simplifies many theoretical and practical applications in vector space analysis.