This statement involves vectors $X_1, X_2, X_3$ in a vector space and examines the relationship between the spans of sets of these vectors.

Step-by-step analysis:

1. Span Definitions:

   • The span of a set of vectors is the set of all possible linear combinations of those vectors. $\text{Span}(X_1, X_2, X_3) = \{ a_1 X_1 + a_2 X_2 + a_3 X_3 \mid a_1, a_2, a_3 \in \mathbb{R} \}$ $\text{Span}(Y_1, Y_2) = \{ b_1 Y_1 + b_2 Y_2 \mid b_1, b_2 \in \mathbb{R} \}$
   • In this problem, we are given that: $Y_1 = X_1 + X_2 \quad \text{and} \quad Y_2 = X_3$

2. Relationship between the spans:

   • First, we know that: $\text{Span}(Y_1, Y_2) = \text{Span}(X_1 + X_2, X_3)$ Since $X_1 + X_2$ and $X_3$ are elements of the span of $X_1, X_2, X_3$, it follows that: $\text{Span}(Y_1, Y_2) \subseteq \text{Span}(X_1, X_2, X_3)$
   • The span of $Y_1$ and $Y_2$ is therefore contained within the span of $X_1, X_2, X_3$ because every linear combination of $Y_1$ and $Y_2$ is a linear combination of $X_1, X_2, X_3$.

3. Containment but not equality:

   • To determine if the spans are equal, let's consider the dimensionality:

     • The span of $X_1, X_2, X_3$ will have at most dimension 3 (if $X_1, X_2, X_3$ are linearly independent).
     • The span of $Y_1 = X_1 + X_2$ and $Y_2 = X_3$ is at most 2-dimensional (since we are working with two vectors).

   • In general, since $Y_1$ is a linear combination of $X_1$ and $X_2$, and $Y_2$ is just $X_3$, the set $\{ Y_1, Y_2 \}$ can generate only a subset of the span of $\{ X_1, X_2, X_3 \}$. The dimension of the span of $\{ Y_1, Y_2 \}$ will typically be 2 (if they are linearly independent), while the dimension of the span of $\{ X_1, X_2, X_3 \}$ can be up to 3.

   • Therefore, the span of $\{ Y_1, Y_2 \}$ is contained in the span of $\{ X_1, X_2, X_3 \}$, but it is not equal to it, unless $X_1, X_2$, and $X_3$ satisfy special conditions (such as linear dependence).

Conclusion:

The span of $\{ Y_1, Y_2 \}$ is indeed contained within the span of $\{ X_1, X_2, X_3 \}$, but not equal to it in general. The reasoning lies in the fact that $Y_1$ is a combination of two vectors, reducing the dimensionality of its span compared to the full span of three vectors.

Would you like further details on how the dimension of spans is determined, or do you have any other questions?

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Here are some related questions for deeper exploration:

1. What conditions must $X_1, X_2, X_3$ satisfy for the spans of $Y_1, Y_2$ and $X_1, X_2, X_3$ to be equal?
2. How does the concept of linear independence affect the dimensionality of the span?
3. What happens to the span when $X_1, X_2, X_3$ are linearly dependent?
4. Can we generalize this result to higher dimensions and more vectors?
5. How do changes in the definition of $Y_1$ and $Y_2$ (e.g., adding scalar multiples) affect the span?

Tip: Always check the linear dependence or independence of vectors before determining the dimensionality of their span.