Find value of T with vectors A and B linearly independant

[ep1cly]

Homework Statement

The vectors a, b are linearly independent. For what values of t are = t^2a + b and d = (2t-3)(a-b) linearly independent.


also another similar question

If the vectors a, b , c are linearly independent, show that a-2b-c, 2a+b, and a+b+c are also linearly independent

Homework Equations

The Attempt at a Solution

for first question

i expanded the (2t-3)(a-b) and then grouped terms. Then i tried finding scalar multiples by solving for t in one eqn and plugging into another, however i failed miserably :(

 

[Dick]

If you think of a and b as the basis of the vector space spanned by a and b then the matrix [[t^2,1],[2t-3,-2t+3]] expresses the linear transformation into the space spanned by c and d. If that matrix is nonsingular then c and d are linearly independent. I'm having trouble thinking of a simpler way to phrase it. Does that make sense to you?