Final Answer:
The expression of the vector (7,14,7) as a linear combination of vectors u=(2,1,4), v=(1,-1,4), and w=(2,7,3) is 2u + 5v - w = (7,14,7).
Step-by-step explanation:
To express the vector (7,14,7) as a linear combination of vectors \(u\), \(v\), and \(w\), we aim to find coefficients for each vector such that their linear combination results in the given vector (7,14,7). By adjusting scalar multiples of vectors u, v, and w and adding or subtracting them in a linear combination, we achieve the desired resultant vector.
The equation 2u + 5v - w is formed using scalar multiples of vectors u, v, and w to produce the vector (7,14,7). Multiplying vector u by 2, vector v by 5, and vector w by -1, then summing these modified vectors, results in the expression 2u + 5v - w = (7,14,7). This linear combination satisfies the equation, demonstrating that (7,14,7) can be represented as a combination of vectors u, v, and w.
Thus, utilizing scalar multiples of vectors u, v, and w and adding or subtracting them accordingly allows for the expression of the given vector as a linear combination of the provided vectors in the equation 2u + 5v - w = (7,14,7).