For which values of L does the vector [l,3,-5] belong to span {[1,0,-2],[-3,1,7]?

Question

I'm not sure how to solve this problem. So far I have created the system of equations

$c_1 -3c_2 = l$; $c_2 = 3$; $-2c_1 + 7c_2 = -5$.

However, I cannot figure out how to solve the set of equations without knowing $l$.

Edit: put 5 in title instead of -5

Answer 1

To find the values of L in which the vector $[L, 3, 5]$ is in the span of the vectors $[1, 0, -2]$ and $[-3, 1, 7]$, you must first ask yourself what it means for a vector to be a member of a spanning set.

In general, let $c_1, c_2, ... , c_k$ be a set of vectors. Then $Span${$c_1, c_2, ... , c_k$} is just the set of every possible linear combination of those vectors. In other words, $Span${$c_1, c_2, ... , c_k$} = {$λ_1c_1 + λ_2c_2 + ... + λ_kc_k : λ_1, λ_2, ..., λ_k ∈ℝ$}.

Now in our case, we are talking about $Span${$[1, 0, -,2], [-3, 1, 7]$}. So if the vector $[L, 3, 5]$ is in our spanning set, it must mean that $[L, 3, 5] = x[1, 0, -2] + y[-3, 1, 7]$ for some real numbers x and y. If you ignore the first columns of each vector, you can see that $[3, 5] = x[0, -2] + y[1, 7]$.

Note that y = 3. Thus, it must be that 5 = -2x + 7y implies that 5 = -2x + 7(3). Solve for x, and you get x = 8.

Looking back at our original vector equation, we see that $[L, 3, 5] = x[1, 0, -2] + y[-3, 1, 7]$ must imply that $[L, 3, 5] = 8[1, 0, -2] + 3[-3, 1, 7]$ since we know our values of x and y

Finally, looking at the first column only, we see that L = 8(1) + 3(-3) = -1.

Answer 2

HINT.- (1) First prove that$[1,0,-2]$ and $[-3,1,7]$ are l. i. then the spam has dimension $2$.

(2)$[L,3,5]=\lambda[1,0,-2]+\mu[-3,1,7]\iff\begin{cases}L=\lambda-3\mu\\3=\mu\\5=-2\lambda+7\mu\end{cases}$

which easily implies $L=-1$

Answer 3

Find the determinant of the matrix formed by these vectors. The value at which the determinant vanishes is your desired value for $L$.