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.

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(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 $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$.

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(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.

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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.

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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!