Question

Let R and R³ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of a for which the set of vectors
{[2-3 α], [3 -1 3], [1 −5 7]} 
does not form a basis of R³ is____.

Hint:

To verify if vectors form a basis in \( \mathbb{R}^3 \), calculate the determinant of the associated matrix. A zero determinant indicates linear dependence.

Answer 1

Correct Answer: 5

To determine the value of α for which the vectors {[2, -3, α], [3, -1, 3], [1, -5, 7]} do not form a basis of R³, we need to check when they become linearly dependent. A set of vectors is linearly dependent if the determinant of the matrix formed by these vectors as rows (or columns) is zero. Build the matrix with the vectors as rows: 

2	-3	α
3	-1	3
1	-5	7

Calculate the determinant of the matrix: det = 2((-1)*7 - (-5)*3) - (-3)(3*7 - 1*3) + α(3*(-5) - (-1)*1). Simplify each term: 1. First term: 2 * (−7 + 15) = 2 * 8 = 16. 2. Second term: 3 * (21 − 3) = 3 * 18 = 54. 3. Third term: α * (−15 + 1) = α * (−14) = −14α. Combine: det = 16 + 54 + (−14α). The vectors are linearly dependent when det = 0: 70 - 14α = 0. Solve for α: 14α = 70. α = 5. Verify α is in the range 5,5: Since the range is exactly [5,5], α = 5 is within this range.