Vector in the Span of Two Vectors

Is a given vector $W = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$ in the span of two vectors $U = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$ and $V = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$?

To determine if a given vector is in the span of two other vectors, you need to understand the concepts of linear combinations and span. The span of two vectors is the set of all possible vectors that can be formed by taking linear combinations of these two vectors. In this context, we ask if the vector W can be expressed as a linear combination of the vectors U and V. This involves finding scalars (let's call them a and b) such that when these scalars multiply U and V respectively, their sum equals W. Another way to frame this problem is setting up a system of equations to see if it has a solution that makes W equal to a combination of U and V. Understanding span and linear combinations not only helps in solving this problem but also lays foundation for more advanced topics like basis, dimension, and transformations in vector spaces. Solving such a problem requires setting up the appropriate equations and checking for consistency, which may involve using techniques like row reduction or understanding geometric interpretations of vector spaces.