Question

In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(t)=3 t, \quad g(t)=|t|\)

Short answer

The functions \(f(t)=3t\) and \(g(t)=|t|\) are linearly independent because they cannot be written as a nontrivial linear combination of one another. The only solution for the constants \(c_1\) and \(c_2\) in the equation \(c_1(3t) + c_2|t| = 0\) is the trivial solution where \(c_1=0\) and \(c_2=0\).

Step-by-step solution

Establish whether these functions can be written as a linear combination

To check whether \(f(t)\) can be written as a linear combination of \(g(t)\), we need to find constants \(c_1\) and \(c_2\) such that \(c_1f(t) + c_2g(t) = 0\). Assume such constants exist, then we have: $$c_1(3t) + c_2|t| = 0$$

Solve for constants \(c_1\) and \(c_2\)

Now, we have to examine two cases: \(t \geq 0\) and \(t < 0\). Case 1: \(t \geq 0\) In this case, \(|t|=t\). Then we have: $$c_1(3t) + c_2t = (3c_1 + c_2)t = 0$$ As the above equation must hold for all \(t \geq 0\), we have: $$3c_1 + c_2 = 0$$ Case 2: \(t < 0\) In this case, \(|t|=-t\). Then we have: $$c_1(3t) - c_2t = (3c_1 - c_2)t = 0$$ As the above equation must hold for all \(t < 0\), we have: $$3c_1 - c_2 = 0$$

Solve the system of equations for the constants \(c_1\) and \(c_2\)

Now we have a system of equations: $$3c_1 + c_2 = 0$$ $$3c_1 - c_2 = 0$$ This system has the unique solution \(c_1=\frac{c_2}{3} = 0\), and \(c_2=0\).

Determine if the functions are linearly independent or dependent

Since the only solution to the system of equations is the trivial solution \(c_1=0\) and \(c_2=0\), we can conclude that the functions \(f(t)=3t\) and \(g(t)=|t|\) are linearly independent.