Question

Define by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right)\). For instance, if \({\mathop{\rm p}\nolimits} \left( t \right) = 3 + 5t + 7{t^2}\), then \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}3\\{15}\end{array}} \right)\).

1. Show that \(T\) is a linear transformation. (Hint: For arbitrary polynomials p, q in \({{\mathop{\rm P}\nolimits} _2}\), compute \(T\left( {{\mathop{\rm p}\nolimits} + {\mathop{\rm q}\nolimits} } \right)\) and \(T\left( {c{\mathop{\rm p}\nolimits} } \right)\).)
2. Find a polynomial p in \({{\mathop{\rm P}\nolimits} _2}\) that spans the kernel of \(T\), and describe the range of \(T\).

Short answer

1. It is proved that \(T\) is a linear transformation.
2. The range of \(T\) is all of \({\mathbb{R}^2}\).

Step-by-step solution

State the condition for linear transformation

The conditions forlinear transformation\(T\)are as follows:

1.\(T\left( {{\mathop{\rm u}\nolimits} + {\mathop{\rm v}\nolimits} } \right) = T\left( {\mathop{\rm u}\nolimits} \right) + T\left( {\mathop{\rm v}\nolimits} \right)\) for all \({\mathop{\rm u}\nolimits} ,{\mathop{\rm v}\nolimits} \,\,{\mathop{\rm in}\nolimits} \,\,V\), and

2 \(T\left( {c{\mathop{\rm u}\nolimits} } \right) = cT\left( {\mathop{\rm u}\nolimits} \right)\) for all \({\mathop{\rm u}\nolimits} \,\,\,{\mathop{\rm in}\nolimits} \,\,V\) and all scalar \(c\).

Show that \(T\) is a linear transformation

a)

Suppose \({\mathop{\rm p}\nolimits} \) and \(q\) are arbitrary polynomials in \({{\mathop{\rm P}\nolimits} _2}\). Then

\(\begin{array}{c}T\left( {{\mathop{\rm p}\nolimits} + q} \right) = \left( {\begin{array}{*{20}{c}}{\left( {{\mathop{\rm p}\nolimits} + q} \right)\left( 0 \right)}\\{\left( {{\mathop{\rm p}\nolimits} + q} \right)\left( 1 \right)}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right) + {\mathop{\rm q}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right) + {\mathop{\rm q}\nolimits} \left( 1 \right)}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{{\mathop{\rm q}\nolimits} \left( 0 \right)}\\{{\mathop{\rm q}\nolimits} \left( 1 \right)}\end{array}} \right)\\ = T\left( {\mathop{\rm p}\nolimits} \right) + T\left( {\mathop{\rm q}\nolimits} \right)\end{array}\)

Consider \(c\) is any scalar. Then,

\(\begin{array}{c}T\left( {c{\mathop{\rm p}\nolimits} } \right) = \left( {\begin{array}{*{20}{c}}{\left( {c{\mathop{\rm p}\nolimits} } \right)\left( 0 \right)}\\{\left( {c{\mathop{\rm p}\nolimits} } \right)\left( 1 \right)}\end{array}} \right)\\ = c\left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right)\\ = cT\left( {\mathop{\rm p}\nolimits} \right)\end{array}\)

Therefore, \(T\) is a li\(T:{{\mathop{\rm P}\nolimits} _2} \to {\mathbb{R}^2}\)near transformation.

Thus, it is proved that \(T\) is a linear transformation.

 Determine polynomial \({\mathop{\rm p}\nolimits} \) in \({{\mathop{\rm P}\nolimits} _2}\) that spans the kernel of \(T\) 

b)

Any quadratic polynomial \({\mathop{\rm q}\nolimits} \) that has \({\mathop{\rm q}\nolimits} \left( 0 \right) = 0\) and \({\mathop{\rm q}\nolimits} \left( 1 \right) = 0\) will be included in the kernel of \(T\). Polynomial \({\mathop{\rm q}\nolimits} \) is then a multiple of \({\mathop{\rm p}\nolimits} \left( t \right) = t\left( {t - 1} \right)\).

Describe the range of T

It is given that any vector \(\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right)\) is in \({\mathbb{R}^2}\).

The polynomial \({\mathop{\rm p}\nolimits} = {{\mathop{\rm x}\nolimits} _1} + \left( {{{\mathop{\rm x}\nolimits} _2} - {{\mathop{\rm x}\nolimits} _1}} \right)t\) have values such that \({\mathop{\rm p}\nolimits} \left( 0 \right) = {{\mathop{\rm x}\nolimits} _1}\) and \({\mathop{\rm p}\nolimits} \left( 1 \right) = {{\mathop{\rm x}\nolimits} _2}\).

Therefore, the range of \(T\) is all of \({\mathbb{R}^2}\).