Consider the vectors a1, a2, a3, and b in R2 as shown in the picture at the end of
this document. Answer the following questions and justify your answers.
(a) Is the vector equation
x1a1 + x2a2 + x3a3 = b
consistent or inconsistent? Explain your answer.
(b) Find the reduced row echelon form of the augmented matrix
[ a1 a2 a3 b ] .
(c) Find the solution set of the vector equation
x1a1 + x2a2 + x3a3 = b.
Represent the solution set in parametric vector form.
If you can solve item (b) in the above problem, then the solution of item (c) is clear. However,
one strategy is to use what we learned in Section 1.5 to solve (c) first and then from (c) deduce
what (b) must be.
Problem 4. Let n, p, q ∈ N. Let
a1, . . . , aq , b1, . . . , bp ∈ Rn.
Assume that
a1, . . . , aq ∈ Span{b1, . . . , bp
}.
Prove the following implication: If q > p, then the vectors a1, . . . , aq are linearly dependent.
OVER


To address these questions comprehensively, let’s break them down systematically.

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### (a) Consistency of the vector equation \( x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + x_3 \mathbf{a}_3 = \mathbf{b} \)
This question involves determining whether the given equation has a solution. The equation is consistent if and only if \(\mathbf{b}\) is a linear combination of \(\mathbf{a}_1, \mathbf{a}_2,\) and \(\mathbf{a}_3\). To analyze this:

1. Form the augmented matrix:
   \[
   [ \mathbf{a}_1 \, \mathbf{a}_2 \, \mathbf{a}_3 \, \mathbf{b} ].
   \]

2. Perform Gaussian elimination to reduce the augmented matrix to its **row echelon form (REF)** or **reduced row echelon form (RREF)**. Check whether the last column (associated with \(\mathbf{b}\)) introduces a contradiction (e.g., a row like \([0 \, 0 \, 0 \, c]\), where \(c \neq 0\)).

If no such contradiction occurs, the system is consistent.

---

### (b) Reduced Row Echelon Form of the Augmented Matrix
To find the reduced row echelon form:
1. Combine \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\), and \(\mathbf{b}\) into an augmented matrix \(A\):
   \[
   A = [\mathbf{a}_1 \, \mathbf{a}_2 \, \mathbf{a}_3 \, \mathbf{b}].
   \]

2. Use row operations (swapping, scaling, and adding multiples of rows) to reduce \(A\) to its reduced row echelon form.

The result will reveal the relationship between \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\), and \(\mathbf{b}\), such as whether \(\mathbf{b}\) lies in the span of \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\).

---

### (c) Solution Set in Parametric Vector Form
If the vector equation is consistent, the solution set describes all possible combinations of \(x_1, x_2, x_3\) that satisfy the equation. To express the solution set:
1. Solve the reduced row echelon form derived in part (b).
2. Write the solution in terms of free variables (if applicable) and represent it in **parametric vector form**:
   \[
   \mathbf{x} = \mathbf{x}_p + t_1 \mathbf{v}_1 + t_2 \mathbf{v}_2 + \cdots,
   \]
   where \(\mathbf{x}_p\) is a particular solution and \(\mathbf{v}_1, \mathbf{v}_2, \dots\) are the parametric directions for the solution space.

---

### Problem 4: Proof of Linearly Dependent Vectors
To prove that \(q > p \implies \mathbf{a}_1, \ldots, \mathbf{a}_q\) are linearly dependent under the given assumptions:

1. **Assumption**: \( \mathbf{a}_1, \ldots, \mathbf{a}_q \in \text{Span}\{\mathbf{b}_1, \ldots, \mathbf{b}_p\} \).
   - This implies that each \(\mathbf{a}_i\) can be written as a linear combination of \(\mathbf{b}_1, \ldots, \mathbf{b}_p\):
     \[
     \mathbf{a}_i = c_{i1} \mathbf{b}_1 + c_{i2} \mathbf{b}_2 + \cdots + c_{ip} \mathbf{b}_p, \quad \text{for } i = 1, \ldots, q.
     \]

2. **Linear Independence and Dimensionality**:
   - The vectors \(\mathbf{b}_1, \ldots, \mathbf{b}_p\) form a basis for their span, which has dimension at most \(p\).
   - Since \(q > p\), there are more vectors (\(\mathbf{a}_1, \ldots, \mathbf{a}_q\)) than the dimension of the span.
   - By the **Pigeonhole Principle** (or rank-nullity theorem), these vectors must be linearly dependent.

3. **Conclusion**:
   - Therefore, \(q > p\) implies \(\mathbf{a}_1, \ldots, \mathbf{a}_q\) are linearly dependent.

---

### Request Clarification
If you provide the specific vectors (\(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \mathbf{b}\)) or their matrix representation, I can calculate the row echelon form and solution explicitly. Let me know how you’d like to proceed!

Consider the vectors a1, a2, a3, and b in R2 as shown in the picture at the end of this document. Answer the following questions and justify your answers.
(a) Is the vector equation
x1a1 + x2a2 + x3a3 = b
consistent or inconsistent? Explain your answer.
(b) Find the reduced row echelon form of the augmented matrix
[ a1 a2 a3 b ] .
(c) Find the solution set of the vector equation
x1a1 + x2a2 + x3a3 = b.
Represent the solution set in parametric vector form.
If you can solve item (b) in the above problem, then the solution of item (c) is clear. However,
one strategy is to use what we learned in Section 1.5 to solve (c) first and then from (c) deduce
what (b) must be.
Problem 4. Let n, p, q ∈ N. Let
a1, . . . , aq , b1, . . . , bp ∈ Rn.
Assume that
a1, . . . , aq ∈ Span{b1, . . . , bp}.
Prove the following implication: If q > p, then the vectors a1, . . . , aq are linearly dependent.

Explore how to solve a vector equation in R2, find the reduced row echelon form of an augmented matrix, and determine the solution set in parametric vector form. This problem also explores the implications of linear dependence when the number of vectors exceeds the dimension of the span. Ideal for college-level linear algebra students.

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