Given Information
We are given with the polynomials \[ \mathbf { p } _ { 1 } ( t ) = 1 + t , \mathbf { p } _ { 2 } ( t ) = 1 - t , \text { and } \mathbf { p } _ { 3 } ( t ) = 2 \] We have to find if the polynomials are linearly independent or not.

Step 1:
We have to verify if the set $ \left\{ \mathbf { p } _ { 1 } ( t ) , \mathbf { p } _ { 2 } ( t ) \right\} $ is linearly independent or not

Step 2:
By inspection we can see that \[ \begin{array} { l } { \mathbf { p } _ { 1 } ( t ) + \mathbf { p } _ { 2 } ( t ) - \mathbf { p } _ { 3 } ( t ) = 1 + t + 1 - t - 2 } \\ { \mathbf { p } _ { 1 } ( t ) + \mathbf { p } _ { 2 } ( t ) - \mathbf { p } _ { 3 } ( t ) = 0 } \\ { ( \text { or } ) } \\ { \mathbf { p } _ { 3 } ( t ) = \mathbf { p } _ { 1 } ( t ) + \mathbf { p } _ { 2 } ( t ) } \end{array} \] Hence the set $ \left\{ \mathbf { p } _ { 1 } , \mathbf { p } _ { 2 } , \mathbf { p } _ { 3 } \right\} $ is linearly dependent in $P_3$. Therefore, the linear dependency is

\[ \mathbf { p } _ { 1 } ( t ) + \mathbf { p } _ { 2 } ( t ) - \mathbf { p } _ { 3 } ( t ) = 0 \]

Step 3:
From the above results, the spanning set is reduced to \[ \begin{aligned} H & = \operatorname { Span } \left\{ \mathbf { p } _ { 1 } , \mathbf { p } _ { 2 } , \mathbf { p } _ { 3 } \right\} \\ & = \operatorname { Span } \left\{ \mathbf { p } _ { 1 } , \mathbf { p } _ { 2 } , \mathbf { p } _ { 1 } + \mathbf { p } _ { 2 } \right\} \end{aligned} \] Check if the reduced spanning set is linearly independent or not. \[ \begin{aligned} a \mathbf { p } _ { 1 } + b \mathbf { p } _ { 2 } & = \mathbf { 0 } \\ a ( 1 + t ) + b ( 1 - t ) & = 0 + 0 \cdot t \\ a + b + ( a - b ) t & = 0 + 0 \cdot t \end{aligned} \] So, no nonzero values of $a$ and $b$ can satisfy the above equation, hence the polynomials are linearly independent. Therefore

The basis for the H is $\left\{ \mathbf { p } _ { 1 } , \mathbf { p } _ { 2 } \right\} $