Find the value of k do the set of vectors 𝑣1=(𝑘,1,1 ),   𝑣2=(0,1,1 ),
𝑣3=(𝑘,0,𝑘 ) form a basis of 𝑅3(𝑅)?

To determine if the set of vectors \(\mathbf{v}_1 = (k, 1, 1)\), \(\mathbf{v}_2 = (0, 1, 1)\), and \(\mathbf{v}_3 = (k, 0, k)\) forms a basis of \(\mathbb{R}^3\), we need to check if these vectors are linearly independent. This is done by checking if the determinant of the matrix formed by these vectors is non-zero.

Let's form the matrix \(A\) with \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) as its columns:

\[
A = \begin{pmatrix}
k & 0 & k \\
1 & 1 & 0 \\
1 & 1 & k
\end{pmatrix}
\]

The determinant of \(A\) is given by:

\[
\text{det}(A) = \begin{vmatrix}
k & 0 & k \\
1 & 1 & 0 \\
1 & 1 & k
\end{vmatrix}
\]

We can expand the determinant as follows:

\[
\text{det}(A) = k \cdot \begin{vmatrix} 1 & 0 \\ 1 & k \end{vmatrix} - 0 \cdot \begin{vmatrix} 1 & 0 \\ 1 & k \end{vmatrix} + k \cdot \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix}
\]

Let's calculate these 2x2 determinants:

\[
\begin{vmatrix} 1 & 0 \\ 1 & k \end{vmatrix} = (1)(k) - (0)(1) = k
\]

\[
\begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} = (1)(1) - (1)(1) = 0
\]

Substituting back into the determinant of \(A\):

\[
\text{det}(A) = k \cdot k + k \cdot 0 = k^2
\]

For the vectors to form a basis, the determinant must be non-zero:

\[
k^2 \neq 0
\]

This implies that \(k \neq 0\).

Thus, the vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) form a basis of \(\mathbb{R}^3\) if and only if \(k \neq 0\).

Would you like more details on any part of this solution?

### Related Questions:
1. What happens if \(k = 0\)? 
2. How do you determine if a different set of vectors forms a basis for \(\mathbb{R}^3\)?
3. Can the vectors form a basis if they are linearly dependent?
4. What does it mean for vectors to span \(\mathbb{R}^3\)?
5. How would this problem change if you were working in \(\mathbb{R}^4\) instead?

### Tip:
When checking if vectors form a basis, remember that their determinant should be non-zero, indicating that they are linearly independent and span the space.

Explore whether the vectors (k, 1, 1), (0, 1, 1), and (k, 0, k) form a basis of \(\mathbb{R}^3\) through matrix determinant analysis. Understand the linear independence condition and its implications. Suitable for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation