1 Find the values of x for which the vectors x 2 0t 1 2 3t 1

(1) Find the values of x for which the vectors (x, 2, 0)^t, (1, 2, 3)^t, (1, x, 1)^t are linearly dependent, where t represents the transpose.

(2) Let A=(a₁, a₂, a₃, a₄) be an 4x4 matrix, where a_j are the column vectors of A, Find a sufficient and necessary condition such that [a₁, 2a₂, 3a₃, 4a₄] forms a basis for R⁴? Explain why.

Solution

1) The vectors are said to be linearly dependent if the determinant of the vectors is equal to zero

Det(A) = x(2-3x) + 1(0-2) + 1(6-0)

2x - 3x^2 + 4 = 0

3x^2 - 2x - 4 = 0

x1 = 1.535 and x2 = -0.867

2)

In order to form the basis of R4, the vectors (a1,a2,a3,a4) must be linearly independent i.e. the determinant of 4X4 matrix must not be zero

Hence the only necessary and sufficient condition that the a1,a2,a3 and a4 must be linearly independent