Determine if b is in the span of the other given vectors. If so, express b as a linear combination of the other vectors. (If b cannot be written as a linear combination of the other two vectors, enter DNE in both answer blanks.)
a1 =

−1

3
−1
,    a2 =

−2

−3
6
,    b =

−5

6
3

To determine whether the vector \( \mathbf{b} \) can be written as a linear combination of vectors \( \mathbf{a}_1 \) and \( \mathbf{a}_2 \), we need to solve the equation:

\[
c_1 \mathbf{a}_1 + c_2 \mathbf{a}_2 = \mathbf{b}
\]

Where:
\[
\mathbf{a}_1 = \begin{pmatrix} -1 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{a}_2 = \begin{pmatrix} -2 \\ -3 \\ 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -5 \\ 6 \\ 3 \end{pmatrix}
\]

This gives the following system of equations:

\[
c_1(-1) + c_2(-2) = -5
\]
\[
c_1(3) + c_2(-3) = 6
\]
\[
c_1(-1) + c_2(6) = 3
\]

We now solve this system to determine if \( c_1 \) and \( c_2 \) exist.

### Equation 1:
\[
-c_1 - 2c_2 = -5
\]

This simplifies to:
\[
c_1 + 2c_2 = 5 \quad \text{(Eq 1)}
\]

### Equation 2:
\[
3c_1 - 3c_2 = 6
\]

Dividing the entire equation by 3:
\[
c_1 - c_2 = 2 \quad \text{(Eq 2)}
\]

### Equation 3:
\[
-c_1 + 6c_2 = 3
\]

This simplifies to:
\[
c_1 - 6c_2 = -3 \quad \text{(Eq 3)}
\]

Now, we can solve these equations step by step.

From Eq 2, we have \( c_1 = c_2 + 2 \).

Substitute \( c_1 = c_2 + 2 \) into Eq 1:
\[
(c_2 + 2) + 2c_2 = 5
\]
\[
c_2 + 2 + 2c_2 = 5
\]
\[
3c_2 + 2 = 5
\]
\[
3c_2 = 3
\]
\[
c_2 = 1
\]

Now, substitute \( c_2 = 1 \) back into Eq 2:
\[
c_1 - 1 = 2
\]
\[
c_1 = 3
\]

### Check Solution with Eq 3:
Substitute \( c_1 = 3 \) and \( c_2 = 1 \) into Eq 3:
\[
3 - 6(1) = -3
\]
\[
3 - 6 = -3
\]
This is true, so the solution is valid.

Thus, \( \mathbf{b} \) can be written as a linear combination of \( \mathbf{a}_1 \) and \( \mathbf{a}_2 \):
\[
\mathbf{b} = 3\mathbf{a}_1 + 1\mathbf{a}_2
\]

### Final Answer:
\[
c_1 = 3, \quad c_2 = 1
\]

Do you need more details or have any questions?

Here are 5 relative questions to expand on this concept:
1. What is the geometric interpretation of linear combinations and vector spans?
2. How can you determine if three vectors are linearly dependent or independent?
3. What does it mean for a vector to be in the span of other vectors?
4. How does the row-reduction method help in solving systems of linear equations like this?
5. Can the cross product be used to find relationships between vectors in 3D space?

**Tip**: Always check your solution by substituting the found coefficients back into the original equations to ensure consistency.

Determine if b is in the span of the other given vectors. If so, express b as a linear combination of the other vectors. (If b cannot be written as a linear combination of the other two vectors, enter DNE in both answer blanks.) a1 = -1, 3, -1; a2 = -2, -3, 6; b = -5, 6, 3

Learn how to determine if vector b = (-5, 6, 3) is a linear combination of vectors a1 = (-1, 3, -1) and a2 = (-2, -3, 6). This problem explores key linear algebra concepts, such as vector spaces, span, and solving systems of equations. Suitable for undergraduate students.

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