There's a problem in the book Linear Algebra and its applications.
For what value of $h$ is $y$ a member of $\text{Span}\{v_1,v_2,v_3\}$?
$$v_1 = \begin{bmatrix}1\\-1\\-2\\\end{bmatrix}$$
$$v_2 = \begin{bmatrix}5\\-4\\-7\\\end{bmatrix}$$
$$v_3 = \begin{bmatrix}-3\\1\\0\\\end{bmatrix}$$
$$y = \begin{bmatrix}-4\\3\\h\\\end{bmatrix}$$
Three space vectors (not all coplanar) can be linearly combined to form the entire space. So any $h, y$ can be represented as a space vector. So any $h, y$ can be represented as a space vector. Shouldn't any $h$ vector $y$ be a member of $\text{Span}\{v_1,v_2,v_3\}$?
I can actually see how the answer given by the author works and I can see that $h$ is equal to 5. However, I desire to know what's wrong with my thought?
Your thought is okay, but you have not verified that the three given $v_i$ are "not all coplanar."
In fact $7v_1-2v_2=v_3$, so the span of $v_1,v_2,v_3$ is the same space as the span of $v_1,v_2.$