311.1.3.16 For what value(s) of h if y in the plane spanned

[karush]

$\tiny{311.1.3.16}$

For what value(s) of h if y in the plane spanned by $v_1$ and $v_2$?
$ v_1=\left[\begin{array}{rr} 1&\\0&\\-2 \end{array}\right],
v_2=\left[\begin{array}{rr} 2&\\1&\\7 \end{array}\right],\textit{ and }
y =\left[\begin{array}{rr} h&\\-3&\\-5 \end{array}\right]$

ok I just wanted to get this posted before i have to go... but I am still ? on what spanning means

 

[jvicens]

Hi there,

Spanning in this context means that the vector $y$ can be written as a linear combination of $v_1$ and $v_2$. In other words, there exists some values of $h$ such that when you multiply $v_1$ and $v_2$ by those values and add them together, you get the vector $y$.

To find the value(s) of $h$ that satisfy this, you can set up a system of equations:

$h\cdot v_1 + (-3)\cdot v_2 = y$

This can be rewritten as:

$\left[\begin{array}{rr} 1&\\0&\\-2 \end{array}\right]h + \left[\begin{array}{rr} -6&\\-3&\\-21 \end{array}\right] = \left[\begin{array}{rr} h&\\-3&\\-5 \end{array}\right]$

Solving this system of equations will give you the value(s) of $h$ that make $y$ in the plane spanned by $v_1$ and $v_2$. In this case, the value of $h$ is $-1$.

I hope this helps! Let me know if you have any other questions.