Final answer:
It is true that if vectors x and y are linearly independent and the set {x, y, z} is linearly dependent, then z is in the span of x and y. Linear dependence indicates that z can be expressed as a linear combination of x and y, therefore, z lies within their span.
Step-by-step explanation:
The student's question deals with the concepts of linear independence, linear dependence, and span in the context of vectors. The question specifically asks whether the vector z lies within the span of vectors x and y given that x and y are linearly independent, while the set {x, y, z} is linearly dependent. This scenario implies that z can be written as a linear combination of x and y, because the linear dependence of the set means that one of the vectors can be expressed using the others. Thus, it is true that z is in the span of x and y.
To explain this, let's recall what linear independence and dependence mean. A set of vectors {v1, v2, ..., vn} is said to be linearly independent if no vector in the set can be expressed as a linear combination of the others. If the set is linearly dependent, then at least one vector can be expressed as a linear combination of the others. In our case, since {x, y} are linearly independent, neither vector can be expressed as a linear combination of the other. However, the addition of z to the set makes it linearly dependent, meaning z must be expressible as a linear combination of x and y, thus lying in their span.
An example of such a linear combination could be z = ax + by, where a and b are constants. This implies that any vector in the span of x and y is simply a linear combination of these two vectors, affirming that z must be in their span.