I understand that if a matrix is singular, it has no inverse. If it has nontrivial solutions, it means at least one solutions exists. If it is linearly dependent, it means that for $a_1 \mathbf{v_1}+a_2 \mathbf{v_2} + ... + a_n \mathbf{ v_n} =\mathbf{ 0}$. Not all the $a$'s are 0. (Not all the coefficients of v_k are zero to satisfy the equation. How does singular relate to nontrivial solutions and nontrivial solutions relate to linear dependent?
Let's say you have 3 vectors: $\vec p_1(x)=a_1x^2+b_1x+3$ $\vec p_2(x)=a_2x^2+b_2x+4$ $\vec p_3(x)=a_3x^2+b_3x+99$
We multiply all the stuff with c and get $c_1\vec p_1(x)+c_2\vec p_2(x)+c_3\vec p_3(x)=0$
Then, we make $a_1c_1+a_2c_2+a_3c_3=0$ $b_1c_1+b_2c_2+b_3c_3=0$ $3c_1+4c_2+99c_3=0$
This coefficient matrix can be singular hence there are nontrivial solutions. So, $\vec p_1$, $\vec p_2$ and $\vec p_3$ are linearly dependent.
OR
This coefficient matrix can be nonsingular hence there are trivial solutions. So, $\vec p_1$, $\vec p_2$ and $\vec p_3$ are linearly independent.