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:

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.