Math Problem Statement

If u = [-8, -6, 8], then compute uu^T and u^T u.

Solution

Let's solve the problem step by step. The problem requires us to compute the following:

  1. uuTuu^T: The outer product of the vector uu with itself.
  2. uTuu^T u: The inner product (or dot product) of the vector uu with itself.

Given:

u=[−8−68]u = \begin{bmatrix} -8 \\ -6 \\ 8 \end{bmatrix}

Step 1: Compute uuTuu^T

The outer product is computed as:

\begin{bmatrix} -8 & -6 & 8 \end{bmatrix}$$ $$uu^T = \begin{bmatrix} (-8)(-8) & (-8)(-6) & (-8)(8) \\ (-6)(-8) & (-6)(-6) & (-6)(8) \\ (8)(-8) & (8)(-6) & (8)(8) \end{bmatrix}$$ $$uu^T = \begin{bmatrix} 64 & 48 & -64 \\ 48 & 36 & -48 \\ -64 & -48 & 64 \end{bmatrix}$$ --- #### Step 2: Compute $$u^T u$$ The dot product is computed as: $$u^T u = \begin{bmatrix} -8 & -6 & 8 \end{bmatrix} \begin{bmatrix} -8 \\ -6 \\ 8 \end{bmatrix}$$ $$u^T u = (-8)(-8) + (-6)(-6) + (8)(8)$$ $$u^T u = 64 + 36 + 64 = 164$$ --- ### Final Answers: $$uu^T = \begin{bmatrix} 64 & 48 & -64 \\ 48 & 36 & -48 \\ -64 & -48 & 64 \end{bmatrix}$$ $$u^T u = 164$$ Let me know if you want more explanation or have any further questions. --- Here are 5 related questions: 1. What is the geometric significance of the outer product $$uu^T$$? 2. How does the inner product $$u^T u$$ relate to the magnitude of $$u$$? 3. How would the results change if $$u$$ were normalized to unit length? 4. Can the matrix $$uu^T$$ be used to construct a projection matrix? If so, how? 5. What properties do symmetric matrices like $$uu^T$$ exhibit? **Tip:** The dot product $$u^T u$$ is equal to the square of the magnitude of the vector $$u$$.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Operations

Formulas

Outer product of a vector: uu^T
Inner product of a vector: u^T u

Theorems

Outer product theorem: For any vector u, the outer product uu^T results in a square matrix.
Inner product theorem: For any vector u, the inner product u^T u gives a scalar value representing the sum of the squared elements of u.

Suitable Grade Level

College Level

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