Which of the following functions lie in the span of the...

Which of the following functions lie in the span of the vector-valued functions $$ \mathbf{f}_1(x)=\left(\begin{array}{l} 1 \\ x \end{array}\right), \quad \mathbf{f}_2(x)=\left(\begin{array}{c} x \\ 1 \end{array}\right), \quad \mathbf{f}_3(x)=\left(\begin{array}{c} x \\ 2 x \end{array}\right) ? $$ (a) $\left(\begin{array}{l}2 \\ 1\end{array}\right)$, (b) $\left(\begin{array}{c}1-2 x \\ 1-x\end{array}\right)$, (c) $\left(\begin{array}{c}1-2 x \\ -1-x\end{array}\right)$, (d) $\left(\begin{array}{c}1+x^2 \\ 1-x^2\end{array}\right)$, (e) $\left(\begin{array}{c}2-x \\ 0\end{array}\right)$

Which of the following functions lie in the span of the vector-valued functions
$$
\mathbf{f}_1(x)=\left(\begin{array}{l}
1 \\
x
\end{array}\right), \quad \mathbf{f}_2(x)=\left(\begin{array}{c}
x \\
1
\end{array}\right), \quad \mathbf{f}_3(x)=\left(\begin{array}{c}
x \\
2 x
\end{array}\right) ?
$$
(a) $\left(\begin{array}{l}2 \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{c}1-2 x \\ 1-x\end{array}\right)$,
(c) $\left(\begin{array}{c}1-2 x \\ -1-x\end{array}\right)$,
(d) $\left(\begin{array}{c}1+x^2 \\ 1-x^2\end{array}\right)$,
(e) $\left(\begin{array}{c}2-x \\ 0\end{array}\right)$

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Key Concepts

Function Space

A function space is a collection of functions that can be added together and scaled by numbers (scalars) while still remaining within the space. Evaluating whether a function is in the span of a set of vector-valued functions involves verifying a set of equations, one for each component, ensuring all operations conform to the rules of the space.

Vector-Valued Functions

Vector-valued functions assign a vector (with multiple components) to each input value, such as a function mapping a real number to a two-dimensional vector. To analyze their span, one must examine each coordinate function separately and ensure that the linear combinations match in every component.

Span

The span of a set of functions is the collection of all possible linear combinations of those functions. In linear algebra, determining whether a function belongs to the span asks if there exists a set of scalar coefficients such that when multiplied with the given functions and summed, they yield the target function.

Linear Combination

A linear combination involves multiplying each function in a set by a corresponding scalar and adding the results together. When checking if a function lies in the span of other functions, one forms a linear combination with unknown coefficients and then determines if these coefficients exist that reproduce the function.

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