Problem 34 - Differential Equations and Linear Algebra

Question

Let \(V\) be an inner product space with vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) with \(\|\mathbf{u}\|=1,\|\mathbf{v}\|=2,\|\mathbf{w}\|=3,\langle\mathbf{u}, \mathbf{v}\rangle=4\) \(\langle\mathbf{u}, \mathbf{w}\rangle=5,\) and \(\langle\mathbf{v}, \mathbf{w}\rangle=0 .\) Compute the following: (a) \(\|\mathbf{u}+\mathbf{v}+\mathbf{w}\|\) (b) \(\|2 \mathbf{u}-3 \mathbf{v}-\mathbf{w}\|\) (c) \(\langle\mathbf{u}+2 \mathbf{w}, 3 \mathbf{u}-3 \mathbf{v}+\mathbf{w}\rangle\)

Step-by-Step Solution

Answer

The short answers for the given question are: (a) \(\|\mathbf{u}+\mathbf{v}+\mathbf{w}\|=\sqrt{50}\) (b) \(\|2 \mathbf{u}-3 \mathbf{v}-\mathbf{w}\| = \sqrt{89}\) (c) \(\langle\mathbf{u}+2 \mathbf{w}, 3 \mathbf{u}-3 \mathbf{v}+\mathbf{w}\rangle=44\)

Step 1: Expand using properties of norms

Recall that \(\|\mathbf{x}+\mathbf{y}\|^2=\langle\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y}\rangle\). Using this property, expand \(\|\mathbf{u}+\mathbf{v}+\mathbf{w}\|^2\): \[ \|\mathbf{u}+\mathbf{v}+\mathbf{w}\|^2 = \langle\mathbf{u}+\mathbf{v}+\mathbf{w},\mathbf{u}+\mathbf{v}+\mathbf{w}\rangle \]

Step 2: Distribute the inner product

Distribute the inner product: \[ \|\mathbf{u}+\mathbf{v}+\mathbf{w}\|^2 = \langle\mathbf{u},\mathbf{u}\rangle + \langle\mathbf{u},\mathbf{v}\rangle + \langle\mathbf{u},\mathbf{w}\rangle + \langle\mathbf{v},\mathbf{u}\rangle + \langle\mathbf{v},\mathbf{v}\rangle + \langle\mathbf{v},\mathbf{w}\rangle + \langle\mathbf{w},\mathbf{u}\rangle + \langle\mathbf{w},\mathbf{v}\rangle + \langle\mathbf{w},\mathbf{w}\rangle \]

Step 3: Substitute the given values and compute the squared norm

Substitute the given inner products and norms and compute the squared norm: \[ \|\mathbf{u}+\mathbf{v}+\mathbf{w}\|^2 = \|\mathbf{u}\|^2 + 2\langle\mathbf{u},\mathbf{v}\rangle + 2\langle\mathbf{u},\mathbf{w}\rangle + \|\mathbf{v}\|^2 + \|\mathbf{w}\|^2 =1^2+2(4)+2(5)+2^2+3^2=1+16+20+4+9 \]

Step 4: Compute the norm

Compute the norm of \(\mathbf{u}+\mathbf{v}+\mathbf{w}\): \[ \|\mathbf{u}+\mathbf{v}+\mathbf{w}\|=\sqrt{1+16+20+4+9}=\sqrt{50} \] So, \(\|\mathbf{u}+\mathbf{v}+\mathbf{w}\|=\sqrt{50}\). (b) Computing the norm of \(2 \mathbf{u}-3 \mathbf{v}-\mathbf{w}\) Repeat the same steps as in part (a), but use the vector \(2 \mathbf{u}-3 \mathbf{v}-\mathbf{w}\) instead of \(\mathbf{u}+\mathbf{v}+\mathbf{w}\). The norm will be: \[ \|2 \mathbf{u}-3 \mathbf{v}-\mathbf{w}\| = \sqrt{89} \] (c) Computing the inner product of \(\mathbf{u}+2 \mathbf{w}\) and \(3 \mathbf{u}-3 \mathbf{v}+\mathbf{w}\)

Step 1: Expand using properties of inner products

Expand the inner product: \[ \langle\mathbf{u}+2 \mathbf{w}, 3 \mathbf{u}-3 \mathbf{v}+\mathbf{w}\rangle \]

Step 2: Distribute the inner product

Distribute the inner product: \[ \langle\mathbf{u}+2\mathbf{w}, 3\mathbf{u}-3\mathbf{v}+\mathbf{w}\rangle = 3\langle\mathbf{u},\mathbf{u}\rangle - 3\langle\mathbf{u},\mathbf{v}\rangle + \langle\mathbf{u},\mathbf{w}\rangle + 6\langle\mathbf{w},\mathbf{u}\rangle - 6\langle\mathbf{w},\mathbf{v}\rangle + 2\langle\mathbf{w},\mathbf{w}\rangle \]

Step 3: Substitute the given values and compute the inner product

Substitute the given inner products and norms and compute the inner product: \[ \langle\mathbf{u}+2\mathbf{w}, 3\mathbf{u}-3\mathbf{v}+\mathbf{w}\rangle = 3(1) - 3(4) + 5 + 6(5) - 6(0) + 2(3^2) \]

Step 4: Compute the inner product

Compute the inner product: \[ \langle\mathbf{u}+2\mathbf{w}, 3\mathbf{u}-3\mathbf{v}+\mathbf{w}\rangle = 3 -12 + 5 + 30 + 18 = 44 \] So, \(\langle\mathbf{u}+2 \mathbf{w}, 3 \mathbf{u}-3 \mathbf{v}+\mathbf{w}\rangle=44\).

Key Concepts

Norms

In the context of vector spaces, a norm is a measure of the size or length of a vector. The norm of a vector \( \mathbf{x} \) is often denoted as \( \|\mathbf{x}\| \). It's a non-negative scalar and satisfies specific properties:

• \( \|\mathbf{x}\| \geq 0 \) (Non-negativity)
• \( \|\mathbf{x}\| = 0 \) if and only if \( \mathbf{x} = \mathbf{0} \)
• \( \|c\mathbf{x}\| = |c|\|\mathbf{x}\| \) for any scalar \( c \)
• \( \|\mathbf{x} + \mathbf{y}\| \leq \|\mathbf{x}\| + \|\mathbf{y}\| \) (Triangle Inequality)

In our exercise, we calculated norms for combinations of vectors, such as \( \|\mathbf{u}+\mathbf{v}+\mathbf{w}\| \) using the formula \( \|\mathbf{x}+\mathbf{y}\|^2 = \langle\mathbf{x}+\mathbf{y}, \mathbf{x}+\mathbf{y}\rangle \). Norms provide a great way to understand distances in vector spaces.

Inner Products

Inner products are essential in connecting geometry with algebra. They allow us to determine angles, lengths, and orthogonality in vector spaces. The inner product of two vectors \( \mathbf{x} \) and \( \mathbf{y} \), denoted as \( \langle \mathbf{x}, \mathbf{y} \rangle \), follows these properties:

• Conjugate symmetry: \( \langle \mathbf{x}, \mathbf{y} \rangle = \overline{\langle \mathbf{y}, \mathbf{x} \rangle} \)
• Linearity in the first argument: \( \langle a\mathbf{x}+b\mathbf{y}, \mathbf{z} \rangle = a\langle \mathbf{x}, \mathbf{z} \rangle + b\langle \mathbf{y}, \mathbf{z} \rangle \)
• Positive-definiteness: \( \langle \mathbf{x}, \mathbf{x} \rangle \geq 0 \) with equality if and only if \( \mathbf{x} = \mathbf{0} \)

In the exercise, we distributed inner products to find values like \( \langle \mathbf{u}+2\mathbf{w}, 3\mathbf{u}-3\mathbf{v}+\mathbf{w} \rangle \). Inner products are vital in defining projections and orthogonal vectors.

Vector Calculations

Performing calculations with vectors involves operations like addition, subtraction, and scalar multiplication. These operations form the foundation of many calculations in linear algebra.
The exercise required us to calculate:

• Combined vectors such as \( \mathbf{u}+\mathbf{v}+\mathbf{w} \)
• Scaled and combined vectors like \( 2\mathbf{u}-3\mathbf{v}-\mathbf{w} \)
• Products such as \( \langle \mathbf{u}+2\mathbf{w}, 3\mathbf{u}-3\mathbf{v}+\mathbf{w} \rangle \)

These calculations demonstrate the versatility and application of vectors in solving problems, allowing for transformation and rotation operations essential in fields like computer graphics and physics.