Question

Find (a) a basis for and (b) the dimension of the solution space of the 

homogeneous system of linear equations.

1.

x−2y +3z= 0

−3𝑥x +6𝑦y -9z = 0

Answer 1

Let us write the system of equations in matrix form $A \bold x = 0$, where $A = \begin{pmatrix} 1 & -2 & 3\\ -3 & 6 & -9 \end{pmatrix}$ and $\bold x = (x, y, z)^T$.

Search for general solution of this system using Gauss method (transforming matrix to row reduced form): $\begin{pmatrix} 1 & -2 & 3\\ -3 & 6 & 9 \end{pmatrix} \sim (II \rightarrow II + 3 I) \sim \begin{pmatrix} 1 & -2 & 3\\ 0 & 0 & 0 \end{pmatrix}$. As one can see, second row vanishes, so the rank of matrix $A$ is 1.

Therefore, general solution to the system can be written as:$\bold x = (2 C_1 - 3 C_2; C_1; C_2)^T = C_1 (2; 1; 0)^T + C_2(-3; 0; 1)^T$.

a) Basis of solution space consists of two vectors: $(2; 1; 0)^T$ and $(-3; 0; 1)^T$.

b) Dimension of the solution space is $2$.