Question

Determine which of the sets of vectors is linearly independent. A: The set where p 1(t) = 1, p 2(t) = t 2, p 3(t) = 1 + 5t B: The set where p 1(t) = t, p 2(t) = t 2, p 3(t) = 2t + 5t 2 C: The set where p 1(t) = 1, p 2(t) = t 2, p 3(t) = 1 + 5t + t 2 A and C C only B only A only all of them

Answer 1

Answer:

(A)A and C

Step-by-step explanation:

In each case, represent each of the $p_i$ as a column vector where  each row corresponds to the constant term, coefficient of t and $t^2$ respectively.

A= The set where $p_1(t)=1$ $p_2(t)=t^2$ $p_3(t)=1+5t$

$A=\left[\begin{array}{ccc}1&0&1\\0&0&5\\0&1&0\end{array}\right]$

$|A|=\left|\begin{array}{ccc}1&0&1\\0&0&5\\0&1&0\end{array}\right|=-5$

B: The set where $p_1(t)=t, p_2(t)=t^2, p_3(t)=2t+5t^2$

$B=\left[\begin{array}{ccc}0&0&0\\1&0&2\\0&1&5\end{array}\right]\\|B|=0$

C: The set where $p_1(t)=1, p_2(t)=t^2, p_3(t)=1+5t+t^2$

$C=\left[\begin{array}{ccc}1&0&1\\0&0&5\\0&1&1\end{array}\right]\\|C|=-5$

Since the determinants of A and C are not 0, the set of vectors in A and C are linearly independent.