Linear Dependence of Vectors in R^4
The given task is to find all pairs of numbers (a, b) with a, b in the set of real numbers R, such that the vectors x = (1, a, 3, 4) and y = (2, 3, b, 8) are linearly dependent in R^4.
Two vectors x and y in R^4 are linearly dependent if there is a non-zero scalar c such that x = cy or y = cx, or equivalently, if at least one of the vectors can be written as a scalar multiple of the other. This implies that the corresponding components of the vectors must be proportional to each other.
For vectors x and y to be linearly dependent:
1 * c = 2 → c = 2, this defines the scalar multiple;
a * c = 3 → a * 2 = 3 → a = 3/2 → a = 1.5;
3 * c = b → 3 * 2 = b → b = 6;
4 * c = 8 → 4 * 2 = 8, this is true for c = 2.
The system holds true for c = 2, so we find that a = 1.5 and b = 6 are the values that make the two vectors linearly dependent. Therefore, the pair of numbers (a, b) that makes x and y linearly dependent is (1.5, 6).
