Determine some properties of the vectors A = (1,1,1,0), B = (0,1,1,1), C = (1,1,0,0)

Consider the three vectors in $\mathbb{R}^4$ ,

$A = (1,1,1,0), \quad B = (0,1,1,1), \quad C = (1,1,0,0).$

Determine whether $A,B,C$ are linearly independent or linearly dependent.
Find a nonzero vector $D$ such that $A,B,C,D$ are dependent.
Find a nonzero vector $E$ such that $A,B,C,E$ are independent.
Using the vector $E$ obtained in part (c) express the vector $X = (1,2,3,4)$ as a linear combination of $A,B,C,E$ .

We have
$\begin{align*} xA + yB + zC = O && \implies && x+z &=0 \\ &&&& x+y+z &= 0 \\ &&&& x+y &= 0 \\ &&&& y &= 0. \end{align*}$

But this implies $x = y = z = 0$ . Hence, $A,B,C$ are linearly independent.
Let $D = A = (1,1,1,0)$ . Then, letting $c_1 = 1$ , $c_2 = c_3 = 0$ , and $c_4 = -1$ we have
$c_1 A + c_2 B + c_3 C + c_4 D = (0,0,0,0).$

Hence, $A,B,C,D$ are linearly dependent.
Let $E = (0,0,0,1)$ . Then
$\begin{align*} E = (0,0,0,1) && \implies && c_1 A + c_2 B + c_3 C + c_4 E &= (0,0,0,0) \\ && \implies && c_1 + c_3 &= 0 \\ &&&& c_1 + c_2 + c_4 &= 0 \\ &&&& c_1 + c_2 &= 0 \\ &&&& c_2 + c_4 &= 0. \end{align*}$

These equations implies $c_1 = c_2 = c_3 = c_4 = 0$ . Hence, $A,B,C,E$ are linearly independent.
So we compute
$\begin{align*} X = c_1 A + c_2 B + c_3 C + c_4 E && \implies && c_1 + c_3 &= 1 \\ && && c_1 + c_2 + c_3 &=2 \\ && && c_1 + c_2 &= 3 \\ && && c_2 + c_4 &= 4. \end{align*}$

Since $c_1 + c_2 = 3$ (from the third equation), the second implies $c_3 = -1$ . Then the first equation implies $c_1 = 2$ . So the third equation implies $c_2 = 1$ , and finally the fourth equation implies $c+4 = 3$ . Hence,

$X = 2A + B - C + 3E.$