Problem 7 - Differential Equations and Linear Algebra

Question

If \(\mathbf{v}=(6-3 i, 4,-2+5 i, 3 i)\) and \(\mathbf{w}=(i, 2 i, 3 i, 4 i)\) use the standard inner product in \(\mathbb{C}^{4}\) to determine, \(\langle\mathbf{v}, \mathbf{w}\rangle,\|\mathbf{v}\|,\) and \(\|\mathbf{w}\|\)

Step-by-Step Solution

Answer

The inner product \(\langle\mathbf{v}, \mathbf{w}\rangle = -33\), the norm of vector \(\mathbf{v}\) is \(\|\mathbf{v}\| = \sqrt{99}\), and the norm of vector \(\mathbf{w}\) is \(\|\mathbf{w}\| = \sqrt{30}\).

Step 1: Find the inner product of the vectors

To find the inner product of the two vectors, we can use the formula \[ \langle\mathbf{v}, \mathbf{w}\rangle = \sum_{i=1}^{4} v_i \overline{w_i} \] where \(v_i\) are the components of vector \(\mathbf{v}\) and \(w_i\) are the components of vector \(\mathbf{w}\). Substituting the given values, we get: \[ \langle\mathbf{v}, \mathbf{w}\rangle = (6-3i)\overline{(i)} + (4)\overline{(2i)} + (-2+5i)\overline{(3i)} + (3i)\overline{(4i)} \]

Step 2: Compute the conjugates of the components of \(\mathbf{w}\)

Now, we will compute the conjugates of each component of \(\mathbf{w}\): \[ \overline{i} = -i \\ \overline{2i} = -2i \\ \overline{3i} = -3i \\ \overline{4i} = -4i \]

Step 3: Multiply the components and sum them up

Now, multiply the components of \(\mathbf{v}\) and the conjugates of \(\mathbf{w}\), and sum up the results: \[ \langle\mathbf{v}, \mathbf{w}\rangle = (6-3i)(-i) + 4(-2i) + (-2+5i)(-3i) + (3i)(-4i) \\ \langle\mathbf{v}, \mathbf{w}\rangle = 3 + 6i - 8i - 15 - 6i + 9i^2 + 12i - 12 \\ \langle\mathbf{v}, \mathbf{w}\rangle = 3 - 27 + 9(-1) = -24-9 = -33 \] So, the inner product \(\langle\mathbf{v}, \mathbf{w}\rangle = -33\).

Step 4: Compute the norms of the vectors

Now, we will find the norms of the vectors using the formula for the norm: For \(\mathbf{v}\): \[ \|\mathbf{v}\| = \sqrt{\langle\mathbf{v}, \mathbf{v}\rangle} = \sqrt{(6-3i)(6+3i) + (4)(4) + (-2+5i)(-2-5i) + (3i)(-3i)} \\ \|\mathbf{v}\| = \sqrt{45 + 16 + 29 + 9} = \sqrt{99} \] For \(\mathbf{w}\): \[ \|\mathbf{w}\| = \sqrt{\langle\mathbf{w}, \mathbf{w}\rangle} = \sqrt{(i)(-i) + (2i)(-2i) + (3i)(-3i) + (4i)(-4i)} \\ \|\mathbf{w}\| = \sqrt{1 + 4 + 9 + 16} = \sqrt{30} \] So the norms of the vectors are \(\|\mathbf{v}\| = \sqrt{99}\) and \(\|\mathbf{w}\| = \sqrt{30}\).

Key Concepts

Vector Norms

A vector norm is a measure of the size or length of a vector. In the context of complex vectors, the norm incorporates both the real and imaginary components. To compute the norm of a vector in the complex space, we use the formula:

• \( \|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v}\rangle} \)

This involves finding the inner product of the vector with itself, then taking the square root. In practice:

• For a vector \( \mathbf{v} = (v_1, v_2, \ldots, v_n) \), the norm is \( \|\mathbf{v}\| = \sqrt{\sum_{i=1}^{n} v_i \overline{v_i}} \)

For example, the vector \( \mathbf{v} = (6 - 3i, 4, -2 + 5i, 3i) \) has a norm:

• \( \|\mathbf{v}\| = \sqrt{(6-3i)(6+3i) + (4)(4) + (-2+5i)(-2-5i) + (3i)(-3i)} \)
• Calculating this, we find \( \|\mathbf{v}\| = \sqrt{99} \).

The norm provides crucial insights into the magnitude of the vector, which is essential for various mathematical, engineering, and physics applications.

Complex Conjugate

The complex conjugate of a complex number changes the sign of the imaginary part. If you have a complex number \( z = a + bi \), its complex conjugate is \( \overline{z} = a - bi \).

For vectors with complex numbers, each component of the vector may have its own complex conjugate. This operation is straightforward but vital for computing inner products in a complex vector space.

• The vector \( \mathbf{w} = (i, 2i, 3i, 4i) \) yields conjugates:
• \( \overline{i} = -i, \overline{2i} = -2i, \overline{3i} = -3i, \overline{4i} = -4i \)

The conjugates are integral for the calculation of inner products because they ensure the expression remains real-valued. By multiplying a component of a vector by the complex conjugate of another, any imaginary parts cancel out.

Mathematical Proofs

Mathematical proofs in linear algebra, especially when dealing with complex vectors, help verify the properties and results that we derive. Proofs involve logical reasoning and steps to establish the truth of a given statement.

With complex inner products, we focus on proving properties such as symmetry and linearity:

• **Symmetry**: \( \langle \mathbf{v}, \mathbf{w} \rangle = \overline{\langle \mathbf{w}, \mathbf{v} \rangle} \)
• **Linearity**: \( \langle a\mathbf{v}, \mathbf{w} \rangle = a \langle \mathbf{v}, \mathbf{w} \rangle \)

To demonstrate these, one might go through the process of explicitly showing calculations like:

• Calculating the inner product \( \langle \mathbf{v}, \mathbf{w} \rangle \) by summing \( v_i \overline{w_i} \).
• Using distributive laws to simplify the result and reach the conclusion of the computed product.

Proofs cement the understanding of the complex vector space and ensure calculations adhere to expected mathematical laws.