Let's go through the exercises and their requirements step-by-step.

Exercise 1

1. Given: $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ are linearly independent, and $A \in \mathbb{R}^{n \times n}$ is an invertible matrix.

   • Objective: Show that $\{A\mathbf{u}, A\mathbf{v}, A\mathbf{w}\}$ are linearly independent.

   Solution Approach:

   • Start by recalling that since $A$ is invertible, it preserves linear independence.
   • Use the property that for an invertible matrix $A$, if $\{ \mathbf{u}, \mathbf{v}, \mathbf{w} \}$ are linearly independent, then $\{A\mathbf{u}, A\mathbf{v}, A\mathbf{w}\}$ will also be linearly independent.

2. Given: $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ are linearly independent.

   • Objective: Show that $\text{span}(\mathbf{u}, \mathbf{v}) \neq \text{span}(\mathbf{u}, \mathbf{v}, \mathbf{w})$.

   Solution Approach:

   • Since $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are linearly independent, $\mathbf{w}$ does not lie in the span of $\{\mathbf{u}, \mathbf{v}\}$.
   • Therefore, $\text{span}(\mathbf{u}, \mathbf{v}, \mathbf{w})$ is a three-dimensional space, while $\text{span}(\mathbf{u}, \mathbf{v})$ is two-dimensional, making the spans unequal.

Exercise 2

Let $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ (with $n \geq 3$) be three normalized vectors.

1. (a) (1pt) Show that $\mathbf{u} \cdot \mathbf{w} \geq -1$.

   Solution Approach:

   • Since both $\mathbf{u}$ and $\mathbf{w}$ are normalized vectors, their dot product is bounded by the Cauchy-Schwarz inequality.
   • Thus, $-1 \leq \mathbf{u} \cdot \mathbf{w} \leq 1$, so $\mathbf{u} \cdot \mathbf{w} \geq -1$.

2. (b) (3pt) Now suppose that $\mathbf{u} \cdot \mathbf{v} = -\frac{1}{2}$ and $\mathbf{u} \cdot \mathbf{w} = -\frac{1}{3}$. Show that $\mathbf{v} \cdot \mathbf{w} \geq -\frac{2}{3}$.

   Solution Approach:

   • Apply the concept of angle relationships or geometric constraints based on the given dot products.
   • Use the triangle inequality in terms of angles between these vectors to find an inequality that includes $\mathbf{v} \cdot \mathbf{w}$ in relation to $\mathbf{u} \cdot \mathbf{v}$ and $\mathbf{u} \cdot \mathbf{w}$.

3. (c) (1pt) Additionally, suppose that $\mathbf{w} \cdot \mathbf{w} = 0$. Complete an admissible triple $\{\mathbf{u}, \mathbf{v}\} = \mathbf{e}_1, \mathbf{w} = \mathbf{e}_j$ satisfying all the relations.

   Solution Approach:

   • This part seems to involve finding vectors that form a basis while satisfying the given conditions.
   • Use standard basis vectors like $\mathbf{e}_1$, $\mathbf{e}_j$, etc., to represent an orthonormal system that adheres to the given dot product constraints.

Would you like detailed solutions for any part, or have any specific questions?

Follow-Up Questions

1. What is the significance of an invertible matrix preserving linear independence in a vector space?
2. Why does the span of three linearly independent vectors differ from that of two?
3. How does the Cauchy-Schwarz inequality apply to normalized vectors?
4. What is the impact of specific dot products in determining angles between vectors?
5. How would changing one of the conditions affect the solution?

Tip

Always check if the matrix is invertible when dealing with transformations and linear independence; invertibility often preserves independence properties.