For the subspace $\left\{ \left( a,b,c\right) :a-3b+c=0,b-2c=0, 2b-c=0\right\}$, find a basis

Question

• For the subspace $\left\{ \left( a,b,c\right) :a-3b+c=0,b-2c=0, 2b-c=0\right\}$, find a basis.

From the set, we get, $a=2c$, $b=c$. My question is:How can I find a basis? Can you help, can you give a hint?

Answer 1

A basis for this subspace is the same as the basis for the solution system to the following matrix equation $$\begin{bmatrix}1&-3&1\\0&1&-2\\0&2&-1\end{bmatrix}\begin{bmatrix}a\\b\\c\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$$

An additional hint: it is very easy to see that this matrix (the coefficient matrix on the left) has nonzero determinant. What does this tell us about the solution system to the above equation?