Determine whether S is a basis for P3. S = {4t - 12,5 +t3,5 +3t,-3t2 +2/3

Answer:Yes , S is a basis for [tex]P_3[/tex].Step-by-step explanation:GivenS=[tex]\left\{4t-12,5+t^3,5+3t,-3t^2+\frac{2}{3}\righ\}[/tex].We can make a matrixLet A=[tex]\begin{bmatrix}-12&4&0&0\\5&0&0&1\\5&3&0&0\\\frac{2}{3}&0&-3&0\end{bmatrix}[/tex]All rows and columns are linearly indepedent and S span [tex]P_3[/tex].Hence, S is a basis of [tex]P_3[/tex]Linearly independent means any row or any column should not combination of any rows or columns.Because  a subset of V with n elements is a basis if and only if it is linearly independent. Basis:- If B is a subset  of a vector space V over a field F .B is basis of V if satisfied the following conditions:1.The elements of B are linearly independent.2.Every element of vector V spanned by the elements of B.