Let uequalsStart 3 By 1 Table 1st Row 1st Column negative 14 2nd Row 1st Column negative 5 3rd Row 1st Column negative 7 EndTable 
  and AequalsStart 3 By 3 Table 1st Row 1st Column 6 2nd Column 8 3rd Column 14 2nd Row 1st Column 1 2nd Column 2 3rd Column 1 3rd Row 1st Column 1 2nd Column 3 3rd Column negative 1 EndTable 
 . Is u in the subset of set of real numbers Rcubed spanned by the columns of​ A? Why or why​ not?
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Part 1
Select the correct choice below and fill in the answer box to complete your choice.
​(Type an integer or decimal for each matrix​ element.)
A.
​No, the reduced echelon form of the augmented matrix is 
  
enter your response here​, which is an inconsistent system.
B.
​Yes, multiplying A by the vector 
  
enter your response here writes u as a linear combination of the columns of A.

To determine if the vector \( u \) is in the subset of \( \mathbb{R}^3 \) spanned by the columns of matrix \( A \), we need to check if there exists a solution to the matrix equation:

\[
A \mathbf{x} = u
\]

where \( A \) is the given \( 3 \times 3 \) matrix, \( u \) is a \( 3 \times 1 \) vector, and \( \mathbf{x} \) is a column vector \( \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \).

### Step 1: Setting Up the Augmented Matrix
We form the augmented matrix \( [A | u] \) by appending vector \( u \) to matrix \( A \):

\[
[A | u] = \begin{bmatrix} 6 & 8 & 14 & | & -14 \\ 1 & 2 & 1 & | & -5 \\ 1 & 3 & -1 & | & -7 \end{bmatrix}
\]

### Step 2: Row Reduction to Reduced Row Echelon Form (RREF)
We need to perform row reduction on this augmented matrix to determine whether it has a solution. I'll carry out these calculations to obtain the RREF.The reduced row echelon form (RREF) of the augmented matrix \( [A | u] \) is:

\[
\begin{bmatrix} 1 & 0 & 5 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}
\]

From this matrix, the last row corresponds to the equation \( 0 = 1 \), which is a contradiction. This indicates that the system is inconsistent.

### Conclusion
The vector \( u \) is **not** in the subset of \( \mathbb{R}^3 \) spanned by the columns of matrix \( A \). Therefore, the correct choice is:

**A. No, the reduced echelon form of the augmented matrix is** 

\[
\begin{bmatrix} 1 & 0 & 5 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix},
\]

**which is an inconsistent system.**

Would you like further details or have any questions?

### Related Questions:
1. How do you determine if a vector is in the span of a set of vectors?
2. What is the significance of an inconsistent system in linear algebra?
3. How do you interpret the reduced row echelon form of an augmented matrix?
4. Can you always express any vector in \( \mathbb{R}^n \) as a linear combination of other vectors in \( \mathbb{R}^n \)?
5. What conditions must the columns of a matrix satisfy for them to span \( \mathbb{R}^3 \)?

### Tip
When dealing with span questions, always check the augmented matrix's RREF to identify if a solution exists!

Let u = Start 3 By 1 Table 1st Row 1st Column negative 14 2nd Row 1st Column negative 5 3rd Row 1st Column negative 7 EndTable and A = Start 3 By 3 Table 1st Row 1st Column 6 2nd Column 8 3rd Column 14 2nd Row 1st Column 1 2nd Column 2 3rd Column 1 3rd Row 1st Column 1 2nd Column 3 3rd Column negative 1 EndTable. Is u in the subset of set of real numbers Rcubed spanned by the columns of A? Why or why not?

Determine if the vector u = [-14, -5, -7] is in the span of the columns of matrix A. This linear algebra problem involves solving a system of equations through row reduction to the reduced row echelon form (RREF). Suitable for undergraduate linear algebra students.

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