The value of t for which (4, 6, 3, t) is a linear combination of (1, 3, - 4, 1), (2, 8, - 5, - 1) and (- 1, - 5, 0, 2) is equal to 13.
How to find a vector as a linear combination of other vectors
In this question we must solve for t such that (4, 6, 3, t) is a linear combination of vectors (1, 3, - 4, 1), (2, 8, - 5, - 1) and (- 1, - 5, 0, 2), which means that non-zero coefficients have to be found by algebraic handling:
(4, 6, 3, t) = α₁ · (1, 3, - 4, 1) + α₂ · (2, 8, - 5, - 1) + α₃ · (- 1, - 5, 0, 2)
Which is equivalent to this system of linear equations:
α₁ + 2 · α₂ - α₃ = 4
3 · α₁ + 8 · α₂ - 5 · α₃ = 6
- 4 · α₁ - 5 · α₂ = 3
α₁ - α₂ + 2 · α₃ - t = 0
Whose solution is (α₁, α₂, α₃, t) = (38, - 31, - 28, 13). The value of t for which (4, 6, 3, t) is a linear combination of (1, 3, - 4, 1), (2, 8, - 5, - 1) and (- 1, - 5, 0, 2) is equal to 13.
To learn more on linear combinations: https://brainly.com/question/9672435
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