Let us find the basis and the dimension of the solution space W W W
{ X 1 + 2 X 2 − 2 X 3 + 2 X 4 − X 5 = 0 X 1 + 2 X 2 − X 3 + 3 X 4 − 2 X 5 = 0 2 X 1 + 4 X 2 − 7 X 3 + X 4 − X 5 = 0 \begin{cases}
X_1 + 2X_2 - 2X_3 + 2X_4 - X_5 = 0\\
X_1 + 2X_2 - X_3 + 3X_4 - 2X_5 = 0\\
2X_1 + 4X_2 - 7X_3 +X_4 - X_5 = 0
\end{cases} ⎩ ⎨ ⎧ X 1 + 2 X 2 − 2 X 3 + 2 X 4 − X 5 = 0 X 1 + 2 X 2 − X 3 + 3 X 4 − 2 X 5 = 0 2 X 1 + 4 X 2 − 7 X 3 + X 4 − X 5 = 0
After substracting from the second equation the first equation, and after multiplying the first equation by − 2 -2 − 2
{ X 1 + 2 X 2 − 2 X 3 + 2 X 4 − X 5 = 0 X 3 + X 4 − X 5 = 0 − 3 X 3 − 3 X 4 + X 5 = 0 \begin{cases}
X_1 + 2X_2 - 2X_3 + 2X_4 - X_5 = 0\\
X_3 + X_4 - X_5 = 0\\
-3X_3 -3X_4 +X_5 = 0
\end{cases} ⎩ ⎨ ⎧ X 1 + 2 X 2 − 2 X 3 + 2 X 4 − X 5 = 0 X 3 + X 4 − X 5 = 0 − 3 X 3 − 3 X 4 + X 5 = 0
Let us multiply the second equation by 3 and add to the third equation:
{ X 1 + 2 X 2 − 2 X 3 + 2 X 4 − X 5 = 0 X 3 + X 4 − X 5 = 0 − 2 X 5 = 0 \begin{cases}
X_1 + 2X_2 - 2X_3 + 2X_4 - X_5 = 0\\
X_3 + X_4 - X_5 = 0\\
-2X_5 = 0
\end{cases} ⎩ ⎨ ⎧ X 1 + 2 X 2 − 2 X 3 + 2 X 4 − X 5 = 0 X 3 + X 4 − X 5 = 0 − 2 X 5 = 0
It follows that
{ X 1 = − 2 X 2 + 2 X 3 − 2 X 4 X 3 = − X 4 X 5 = 0 \begin{cases}
X_1 =- 2X_2 +2X_3 -2X_4 \\
X_3 =- X_4\\
X_5 = 0
\end{cases} ⎩ ⎨ ⎧ X 1 = − 2 X 2 + 2 X 3 − 2 X 4 X 3 = − X 4 X 5 = 0
and hence
{ X 1 = − 2 X 2 − 4 X 4 X 3 = − X 4 X 5 = 0 \begin{cases}
X_1 =- 2X_2 -4X_4 \\
X_3 =- X_4\\
X_5 = 0
\end{cases} ⎩ ⎨ ⎧ X 1 = − 2 X 2 − 4 X 4 X 3 = − X 4 X 5 = 0
We conclude that
W = { ( X 1 , X 2 , X 3 , X 4 , X 5 ) ∈ R 5 : X 1 = − 2 X 2 − 4 X 4 , X 3 = − X 4 , X 5 = 0 } = { ( − 2 X 2 − 4 X 4 , X 2 , − X 4 , X 4 , 0 ) ∈ R 5 : X 2 , X 4 ∈ R } = { X 2 ( − 2 , 1 , 0 , 0 , 0 ) + X 4 ( − 4 , 0 , − 1 , 1 , 0 ) ∈ R 5 : X 2 , X 4 ∈ R } . W=\{(X_1,X_2,X_3,X_4,X_5)\in\R^5: X_1 =- 2X_2 -4X_4,\ X_3 =- X_4, \ X_5 = 0\}\\
=\{(- 2X_2 -4X_4,X_2,-X_4,X_4,0)\in\R^5: X_2, X_4 \in\R\}\\
=\{X_2(- 2,1,0,0,0)+X_4(-4,0,-1,1,0)\in\R^5: X_2, X_4 \in\R\}. W = {( X 1 , X 2 , X 3 , X 4 , X 5 ) ∈ R 5 : X 1 = − 2 X 2 − 4 X 4 , X 3 = − X 4 , X 5 = 0 } = {( − 2 X 2 − 4 X 4 , X 2 , − X 4 , X 4 , 0 ) ∈ R 5 : X 2 , X 4 ∈ R } = { X 2 ( − 2 , 1 , 0 , 0 , 0 ) + X 4 ( − 4 , 0 , − 1 , 1 , 0 ) ∈ R 5 : X 2 , X 4 ∈ R } .
Therefore, { ( − 2 , 1 , 0 , 0 , 0 ) , ( − 4 , 0 , − 1 , 1 , 0 ) } \{(- 2,1,0,0,0),(-4,0,-1,1,0)\} {( − 2 , 1 , 0 , 0 , 0 ) , ( − 4 , 0 , − 1 , 1 , 0 )} W W W d i m W = 2. dim\ W=2. d im W = 2.
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