In one of the problems in the book we are asked to prove that the set of seven functions

{1,Cos[t],(((Cos[t]))^2),(((Cos[t]))^3),(((Cos[t]))^4),(((Cos[t]))^5),(((Cos[t]))^6)}

is linearly independent. In Mathematica such a set can be formed as follows:

Table[(((Cos[t]))^k),{k,1,6}]

Prepend[Table[(((Cos[t]))^k),{k,1,6}],1]

Above, I am a little paranoid not to use

(((Cos[t]))^0)

instead of 1, since zero to the power zero is not defined. So, formally the expression in the preceding line is not well-defined. Therefore, I use Prepend

Now we want to assign seven values to t. We do it as follows:

Table[t,{t,0,((6 Pi)/7),(( Pi)/7)}]

Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,((6 Pi)/7),(( Pi)/7)}]

The above expression is exactly the matrix that we need to study:

MatrixForm[Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,((6 Pi)/7),(( Pi)/7)}]]

Calculate the determinant:

FullSimplify[Det[Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,((6 Pi)/7),(( Pi)/7)}]]]

Row reduce:

RowReduce[Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,((6 Pi)/7),(( Pi)/7)}]]

We could have used some other values of t. For example

TeXForm[Table[t,{t,0,Pi,(( Pi)/6)}]]

MatrixForm[Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,Pi,(( Pi)/6)}]]

TeXForm[Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,Pi,(( Pi)/6)}]]

Det[Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,Pi,(( Pi)/6)}]]

TeXForm[Det[Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,Pi,(( Pi)/6)}]]]

RowReduce[Table[Prepend[Table[(((Cos[t]))^k),{k,1,6}],1],{t,0,Pi,(( Pi)/6)}]]