Which of the following set of vectors forms a basis of of vector space V3(R)

Question 1Answer

a.
{(1,0,0),(1,1,0),(1,1,1)}

b.
{(1,0,1),(0,1,0),(1,1,1)}

c.
{(1,0,0),(0,1,0),(1,1,0)}

d.
None of these

To determine which of the given sets of vectors forms a basis for the vector space \( V_3(\mathbb{R}) \), we need to check if the vectors in each set are **linearly independent** and if there are exactly **three vectors** (since the dimension of \( V_3(\mathbb{R}) \) is 3). If both conditions are satisfied, the set forms a basis.

### Checking Linearity Independence:
For a set of vectors \(\{v_1, v_2, v_3\}\), the vectors are linearly independent if the only solution to the equation:

\[
c_1 v_1 + c_2 v_2 + c_3 v_3 = 0
\]

is \( c_1 = c_2 = c_3 = 0 \).

This can be checked by forming a matrix with these vectors as rows or columns and checking if the determinant of this matrix is non-zero. If the determinant is non-zero, the vectors are linearly independent.

Let's check each option:

### Option a: \(\{(1,0,0),(1,1,0),(1,1,1)\}\)

The matrix formed by these vectors is:

\[
\begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & 1 & 1 \\
\end{pmatrix}
\]

To check if these vectors are linearly independent, we calculate the determinant of this matrix:

\[
\text{det} = 1\left(1 \cdot 1 - 0 \cdot 1\right) - 0\left(1 \cdot 1 - 0 \cdot 1\right) + 0\left(1 \cdot 1 - 1 \cdot 1\right) = 1 \cdot 1 = 1 \neq 0
\]

Since the determinant is non-zero, the vectors are linearly independent.

### Option b: \(\{(1,0,1),(0,1,0),(1,1,1)\}\)

The matrix formed by these vectors is:

\[
\begin{pmatrix}
1 & 0 & 1 \\
0 & 1 & 0 \\
1 & 1 & 1 \\
\end{pmatrix}
\]

We calculate the determinant:

\[
\text{det} = 1\left(1 \cdot 1 - 0 \cdot 1\right) - 0\left(0 \cdot 1 - 1 \cdot 1\right) + 1\left(0 \cdot 1 - 1 \cdot 1\right) = 1 \cdot 1 - 1 \cdot 1 = 0
\]

Since the determinant is zero, the vectors are **not** linearly independent.

### Option c: \(\{(1,0,0),(0,1,0),(1,1,0)\}\)

The matrix formed by these vectors is:

\[
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
1 & 1 & 0 \\
\end{pmatrix}
\]

We calculate the determinant:

\[
\text{det} = 1\left(1 \cdot 0 - 0 \cdot 1\right) - 0\left(0 \cdot 0 - 0 \cdot 1\right) + 0\left(0 \cdot 1 - 1 \cdot 1\right) = 1 \cdot 0 = 0
\]

Since the determinant is zero, the vectors are **not** linearly independent.

### Conclusion:
The only set of vectors that forms a basis for \( V_3(\mathbb{R}) \) is **Option a**.

So, the correct answer is:

**a. \(\{(1,0,0),(1,1,0),(1,1,1)\}\)**

Would you like more details, or do you have any questions?

Here are some related questions:
1. How do you determine if a set of vectors spans a vector space?
2. What does it mean for vectors to be linearly independent?
3. Can two vectors ever form a basis for \( V_3(\mathbb{R}) \)?
4. How can you check if a set of vectors forms a basis using the rank of a matrix?
5. What is the significance of the determinant in determining linear independence?

**Tip:** If the determinant of a matrix formed by vectors is zero, the vectors are linearly dependent and do not form a basis.

Learn how to determine which set of vectors forms a basis for the vector space V3(R) with a detailed step-by-step solution. This problem involves checking for linear independence and calculating determinants. Suitable for undergraduate students studying linear algebra.

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