Final answer:
The statement is true because a subset of a linearly independent set is also linearly independent, meaning {u1, u2, u3} retains this property if {u1, u2, u3, u4} is linearly independent.
Step-by-step explanation:
The statement If {u1, u2, u3, u4} is linearly independent, then so is {u1, u2, u3} is true. To understand why this is the case, we need to define linear independence. A set of vectors is considered linearly independent if there is no way to write any vector as a linear combination of the others. More formally, if the only solution to the equation c1*u1 + c2*u2 + c3*u3 + ... + cn*un = 0 is c1 = c2 = c3 = ... = cn = 0, then the set is linearly independent.
Now, if a set of four vectors {u1, u2, u3, u4} is linearly independent, this means none of the vectors can be written as a linear combination of the others. When we consider a subset of these vectors, such as {u1, u2, u3}, this property still holds. Since u4 was independent of u1, u2, and u3, removing it does not create a linear dependence amongst u1, u2, and u3. Therefore, {u1, u2, u3} must also be linearly independent.
If we had instead added a vector to {u1, u2, u3} to create {u1, u2, u3, u4}, we would need to check for linear independence again as the additional vector could potentially be a linear combination of the previous ones. However, in this scenario where we know the larger set is already independent, we can confidently state that the subset retains that property.