1 point) The vectors
𝑣⃗ 1=⎡⎣⎢⎢2−20⎤⎦⎥⎥,    𝑣⃗ 2=⎡⎣⎢⎢−156⎤⎦⎥⎥,    𝑣⃗ 3=⎡⎣⎢⎢31𝑘⎤⎦⎥⎥
form a basis for ℝ3
 if and only if 𝑘≠
 

To determine the condition on \( k \) that ensures the vectors \( \mathbf{v}_1 = \begin{bmatrix} 2 \\ -2 \\ 0 \end{bmatrix} \), \( \mathbf{v}_2 = \begin{bmatrix} -1 \\ 5 \\ 6 \end{bmatrix} \), and \( \mathbf{v}_3 = \begin{bmatrix} 3 \\ 1 \\ k \end{bmatrix} \) form a basis for \( \mathbb{R}^3 \), we need to check whether these vectors are **linearly independent**.

### Step 1: Form the matrix
The vectors form a basis if the matrix constructed from these vectors as columns has a non-zero determinant. The matrix is:

\[
A = \begin{bmatrix} 
2 & -1 & 3 \\
-2 & 5 & 1 \\
0 & 6 & k 
\end{bmatrix}
\]

### Step 2: Calculate the determinant of the matrix
The determinant of this 3x3 matrix is:

\[
\text{det}(A) = \begin{vmatrix} 
2 & -1 & 3 \\
-2 & 5 & 1 \\
0 & 6 & k 
\end{vmatrix}
\]

We will expand along the third row since it contains a zero, simplifying the calculation:

\[
\text{det}(A) = 0 \cdot \text{det}\begin{bmatrix} -1 & 3 \\ 5 & 1 \end{bmatrix} - 6 \cdot \text{det}\begin{bmatrix} 2 & 3 \\ -2 & 1 \end{bmatrix} + k \cdot \text{det}\begin{bmatrix} 2 & -1 \\ -2 & 5 \end{bmatrix}
\]

### Step 3: Compute the 2x2 minors
- \(\text{det}\begin{bmatrix} -1 & 3 \\ 5 & 1 \end{bmatrix} = (-1)(1) - (3)(5) = -1 - 15 = -16 \)
- \(\text{det}\begin{bmatrix} 2 & 3 \\ -2 & 1 \end{bmatrix} = (2)(1) - (3)(-2) = 2 + 6 = 8 \)
- \(\text{det}\begin{bmatrix} 2 & -1 \\ -2 & 5 \end{bmatrix} = (2)(5) - (-1)(-2) = 10 - 2 = 8 \)

### Step 4: Substitute and simplify
Now, substitute these values into the determinant formula:

\[
\text{det}(A) = 0 \cdot (-16) - 6 \cdot 8 + k \cdot 8
\]

\[
\text{det}(A) = -48 + 8k
\]

### Step 5: Solve for linear independence
For the vectors to form a basis, the determinant must not be zero:

\[
-48 + 8k \neq 0
\]

Solving for \( k \):

\[
8k \neq 48
\]

\[
k \neq 6
\]

### Final Answer:
The vectors form a basis for \( \mathbb{R}^3 \) if and only if \( k \neq 6 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you compute the determinant of a 3x3 matrix?
2. What does it mean for vectors to be linearly independent?
3. How does the determinant relate to the volume of a parallelepiped in 3D space?
4. What happens if \( k = 6 \) in the context of this problem?
5. Can you find a general formula for the determinant of an arbitrary 3x3 matrix?

**Tip:** Always check the determinant of the matrix when determining whether a set of vectors forms a basis for a vector space!

The vectors 𝑣⃗ 1=⎡⎣⎢⎢2−20⎤⎦⎥⎥, 𝑣⃗ 2=⎡⎣⎢⎢−156⎤⎦⎥⎥, 𝑣⃗ 3=⎡⎣⎢⎢31𝑘⎤⎦⎥⎥ form a basis for ℝ3 if and only if 𝑘≠?

Find the value of k such that the vectors v1 = [2, -2, 0], v2 = [-1, 5, 6], and v3 = [3, 1, k] form a basis for ℝ3. This problem involves linear algebra concepts like determinants and vector spaces. Detailed solution and explanation provided.

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