Questions: Cho hệ u1, u2, u3, u4, u5 trong không gian tuyến tính V. Biểu thức nào sau đây không phải là một tỏ̉ hợp tuyến tính của hệ u1, u2, u4, u5 : A. 2 u1+3 u2+u4+4 u5 B. u1+u2-6 u5 C. u2+u5 D. 3 u1-2 u2+2 u4+3 u5 E. 4 u2+u3+2 u4

Solution Steps

To determine which expression is not a linear combination of the set \(\{u_{1}, u_{2}, u_{4}, u_{5}\}\), we need to check if each expression can be formed using only these vectors. The expression that includes \(u_{3}\) cannot be a linear combination of \(\{u_{1}, u_{2}, u_{4}, u_{5}\}\) because \(u_{3}\) is not part of this set.

We are given the set of vectors \(\{u_{1}, u_{2}, u_{3}, u_{4}, u_{5}\}\) and need to determine which of the provided expressions is not a linear combination of the subset \(\{u_{1}, u_{2}, u_{4}, u_{5}\}\).

We analyze each expression to see if it can be formed using only the vectors from the subset \(\{u_{1}, u_{2}, u_{4}, u_{5}\}\):

• A: \(2u_{1} + 3u_{2} + u_{4} + 4u_{5}\) (valid)
• B: \(u_{1} + u_{2} - 6u_{5}\) (valid)
• C: \(u_{2} + u_{5}\) (valid)
• D: \(3u_{1} - 2u_{2} + 2u_{4} + 3u_{5}\) (valid)
• E: \(4u_{2} + u_{3} + 2u_{4}\) (invalid due to the presence of \(u_{3}\))

From the analysis, we find that expression E includes \(u_{3}\), which is not part of the subset \(\{u_{1}, u_{2}, u_{4}, u_{5}\}\). Therefore, it cannot be expressed as a linear combination of the given vectors.

Final Answer