Questions: Write each vector as a linear combination of the vectors in S. (Use s1 and s2, respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.) S=(1,2,-2),(2,-1,1) (a) z=(-5,-5,5) z= (b) v=(-2,-5,5) y= (c) w=(2,-16,16) w= (d) u=(1,-5,-5) Because= IMPOSSIBLE

Write each vector as a linear combination of the vectors in S. (Use s1 and s2, respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.)
S=(1,2,-2),(2,-1,1)
(a) z=(-5,-5,5)
z=
(b) v=(-2,-5,5)
y=
(c) w=(2,-16,16)
w=
(d) u=(1,-5,-5)
Because= IMPOSSIBLE

Transcript text: Write each vector as a linear combination of the vectors in S. (Use $s_{1}$ and $s_{2}$, respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.) \[ S=\{(1,2,-2),(2,-1,1)\} \] (a) $z=(-5,-5,5)$ \[ z=\square \] (b) $\mathrm{v}=(-2,-5,5)$ \[ \mathbf{y}= \] (c) $w=(2,-16,16)$ \[ w=\square \] (d) $\mathbf{u}=(1,-5,-5)$ \[ \because=\text { IMPOSSIBLE } \]

Solution

Solution Steps

To express each vector as a linear combination of the vectors in set $ S = \{(1,2,-2),(2,-1,1)\} $, we need to find scalars $ a $ and $ b $ such that for a given vector $ v $, $ v = a \cdot (1,2,-2) + b \cdot (2,-1,1) $. This involves solving a system of linear equations for each vector.

Step 1: Define the Problem

We need to express each given vector as a linear combination of the vectors in the set $ S = \{(1, 2, -2), (2, -1, 1)\} $. This means finding scalars $ a $ and $ b $ such that for a given vector $ v $, $ v = a \cdot (1, 2, -2) + b \cdot (2, -1, 1) $.

Step 2: Set Up the System of Equations

For each vector, we set up a system of linear equations based on the components of the vectors:

For vector $ z = (-5, -5, 5) $: \[ \begin{align_} a + 2b &= -5 \\ 2a - b &= -5 \\ -2a + b &= 5 \end{align_} \]
For vector $ v = (-2, -5, 5) $: \[ \begin{align_} a + 2b &= -2 \\ 2a - b &= -5 \\ -2a + b &= 5 \end{align_} \]
For vector $ w = (2, -16, 16) $: \[ \begin{align_} a + 2b &= 2 \\ 2a - b &= -16 \\ -2a + b &= 16 \end{align_} \]

Step 3: Solve the System of Equations

We attempt to solve each system of equations to find the values of $ a $ and $ b $. If the system has no solution, it means the vector cannot be expressed as a linear combination of the vectors in $ S $.

Step 4: Analyze the Results

For each vector, the system of equations was found to be inconsistent, meaning there is no solution for $ a $ and $ b $. Therefore, each vector cannot be expressed as a linear combination of the vectors in $ S $.