# [SOLVED]10.3 Determine if A is in the span B

#### karush

##### Well-known member
Determine if $A=\begin{bmatrix} 1\\3\\2 \end{bmatrix}$ is in the span $B=\left\{\begin{bmatrix} 2\\1\\0 \end{bmatrix} \cdot \begin{bmatrix} 1\\1\\1 \end{bmatrix}\right\}$

ok I added A and B to this for the OP
but from examples it looks like this can be answered by scalors so if

$c_1\begin{bmatrix} 2\\1\\0 \end{bmatrix} + c_2\begin{bmatrix} 1\\1\\1 \end{bmatrix}=\begin{bmatrix} 1\\3\\2 \end{bmatrix}$

#### Olinguito

##### Well-known member
Hi karush .

So you have
$$\begin{eqnarray}2c_1 &+& c_2 &=& 1 \\ c_1 &+& c_2 &=& 3 \\ {} & {} & c_2 &=& 2.\end{eqnarray}$$
If you substitute $c_2=2$ from the last equation into the first two equations, you get two different values for $c_1$. Hence the above set of equations is inconsistent (has no solutions) showing that $\mathbf A\notin\mathrm{span}B$.

#### karush

##### Well-known member
Lets try this one.... if $A= \begin{bmatrix} 1\\3\\2 \end{bmatrix}$ is in the span $B=\left\{\begin{bmatrix} 2\\1\\0 \end{bmatrix} \cdot \begin{bmatrix} 1\\1\\1 \end{bmatrix} \cdot \begin{bmatrix} 0\\1\\1 \end{bmatrix}\right\}$
then
$\begin{array}{rrrrr} 2c_1 &+ c_2 & & =1 \\ c_1 &+ c_2 & +c_3 & =3 \\ & c_2 & +c_3 & =2 \end{array}$
Solving $c_1=1, c_2=−1, c_3=3$
so $A\in\mathrm{span}B$

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#### HallsofIvy

##### Well-known member
MHB Math Helper
You are missing some "+" signs, aren't you?

Yes, the definition of "span" requires that
$2c_1+ c_2= 1$
$c_1+ c_2+ c_3= 3$ and
$c_2+ c_3= 2$