Question

Show that the subset $\left\{\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right)\right\}$of ${\mathit{Q}}^{\mathbf{3}}$is linearly independent over $\mathit{Q}$.

Short answer

Answer:

$\{(1,0,0),(0,1,0),(0,0,1)\}$is linearly independent over$Q$

Step-by-step solution

Step 1:Definition of linearly independent vector.

In a vector space , a set of the vector is said to be linearly dependent if one of the vectors in the set can be written as a linear combination of other, and if the no vectors can be written as a linear combination of others then the vector is said to be linearly independent.

Showing that {(1,0,0),(0,1,0),(0,0,1)} is linearly independent.

The subset $\mathbf{\{}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\}}$of ${\mathit{Q}}^{\mathbf{3}}$over $\mathit{Q}$.

Now, to check its independence.

Let ${\mathit{c}}_{\mathbf{1}}{\mathit{c}}_{\mathbf{2}}\mathbf{,}{\mathit{c}}_{\mathbf{3}}\mathbf{\in }\mathit{Q}$are such that

$$\begin{gathered} c_1\left(1,0,0\right)+c_2\left(0,1,0\right)+c_3\left(0,0,1\right)=\left(0,0,0\right) \\ \left(c_1,c_2,c_3\right)=\left(0,0,0\right) \end{gathered}$$

This implies

$$\begin{gathered} c_1=0 \\ {c}_2=0 \\ {c}_3=0 \end{gathered}$$

Since all the coefficient of element of the given subsets are 0.

Therefore, the subset $\mathbf{\{}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\}}$of ${\mathit{Q}}^{\mathbf{3}}$is linearly independent over$\mathit{Q}$.