Is this vector in the span?

Question

a. Is $(13, -12, 14, 4)^T$ in $\mathrm{Span}\{(-4, 3, 2, -2)^T, (-2, 2, -3, 4)^T, (5, -4, 0, 4)^T\}$?

b. Is $1 + x^2$ in $\mathrm{Span}\{-4 + 3x + 2x^2 - 2x^3, 5 - 4x + 4x^3\}$?

c. Is $(15, -14, 18, 8)$ in $\mathrm{Span} \{(-4, 3, 2, -2), (-2, 2, -3, 4), (5, -4, 0, 4), (3, -3, 5 -2), (13, -12, 14, 4)\}$?

I got "yes" and "no" for the first two and I am having trouble with the last one. Could someone help me out and let me know if I got the others correct?

Answer 1

Set up your equation with arbitrary constants:

$\begin{align}c_1<-4, 3, 2, -2> + c_2<-2, 2, -3, 4> + c_3<5, -4, 0, 4> + c_4<3, -4, 5, -2> + c_5<13,-12, 14, 4> = <15, -14, 18, 8>\end{align}$

Now set up the augmented matrix and row-reduce it (hint: let each vector be a column, with the vector $<15, -14, 18, 8>$ to the right of the augment line). Forgive me for not actually typing out the matrix, as it's a bit of a pain.

After it's row-reduced, you'll see that $c_4$ and $c_5$ are free variables. Algebraically solve for the others, and our final set of constants is:

• $c_1 = 6 - c_4- 4c_5 $
• $c_2 = c_4+2c_5-2$
• $c_3 = 7 - c_4- 5c_5 $
• $c_4 = c_4$
• $c_5 = c_5$

So yes, that vector is in the span of the set of vectors.