1.5 Exercises

20 points · 4 extra credit

Solution Sets of Linear Systems

1. 1. (2 pts) Solve the homogeneous system and write its solution set in parametric vector form:

   $\begin{aligned} x_1 - 2x_2 + 3x_3 &= 0 \\ 2x_1 - 3x_2 + 4x_3 &= 0. \end{aligned}$

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2. 2. (3 pts) Find the solution set of $A\mathbf{x} = \mathbf{0}$ in parametric vector form, where

   $A = \begin{bmatrix} 1 & 2 & -1 & 3 \\ 2 & 4 & -1 & 8 \\ 1 & 2 & 0 & 5 \end{bmatrix}.$

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3. 3. (3 pts) Solve the system below and express its solution in the form $\mathbf{x} = \mathbf{p} + x_2\mathbf{v}$:

   $\begin{aligned} x_1 + 2x_2 + x_3 &= 4 \\ 2x_1 + 4x_2 + x_3 &= 5. \end{aligned}$

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4. 4. (3 pts) For the system in Problem 3, write the solution set of the associated homogeneous system $A\mathbf{x} = \mathbf{0}$, and describe geometrically how the two solution sets are related.

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5. 5. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) A homogeneous system $A\mathbf{x} = \mathbf{0}$ is always consistent.

   (b) If $A\mathbf{x} = \mathbf{0}$ has only the trivial solution, then $A\mathbf{x} = \mathbf{b}$ has a unique solution for every $\mathbf{b}$.

   (c) If $A\mathbf{x} = \mathbf{b}$ is consistent, then its solution set is a translate of the solution set of $A\mathbf{x} = \mathbf{0}$.

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6. 6. (3 pts) Describe geometrically, in $\mathbb{R}^3$, the two solution sets, and state how they differ:

   (a) $\mathbf{x} = t\begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}$

   (b) $\mathbf{x} = \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} + t\begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}$

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7. 7. (3 pts) Prove that every homogeneous linear system with more unknowns than equations has a nontrivial solution.

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8. 8. (4 pts ••) Suppose $A\mathbf{x} = \mathbf{b}$ is consistent, and let $\mathbf{p}$ be one solution. Prove that the solution set of $A\mathbf{x} = \mathbf{b}$ equals $\{\mathbf{p} + \mathbf{v}_h : A\mathbf{v}_h = \mathbf{0}\}$.

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