Problem 1 - Differential Equations and Linear Algebra

Question

Show that the given vector functions are linearly independent on \((-\infty, \infty)\). $$\mathbf{x}_{1}(t)=\left[\begin{array}{r} e^{t} \\ -e^{t} \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{l} e^{t} \\ e^{t} \end{array}\right]$$

Step-by-Step Solution

Answer

The given vector functions \(\mathbf{x}_{1}(t)=\left[\begin{array}{r} e^{t} \\ -e^{t} \end{array}\right]\) and \(\mathbf{x}_{2}(t)=\left[\begin{array}{l} e^{t} \\ e^{t} \end{array}\right]\) are linearly independent on \((-\infty, \infty)\) because the only solution to the system of equations formed by their linear combination equal to the zero vector is when the scalars (weights) \(a = 0\) and \(b = 0\).

Step 1: Write the linear combination - the equation we want to solve for

We want to find scalars (or weights) \(a\) and \(b\) such that: $$a\mathbf{x}_{1}(t) + b\mathbf{x}_{2}(t) = \mathbf{0}$$ where \(\mathbf{0}\) is the zero vector.

Step 2: Rewrite the equation in terms of the given vector functions

Substitute the given vector functions: $$a\left[\begin{array}{r} e^{t} \\\ -e^{t} \end{array}\right] + b\left[\begin{array}{l} e^{t} \\\ e^{t} \end{array}\right] = \left[\begin{array}{c} 0 \\\ 0 \end{array}\right]$$

Step 3: Set up the system of equations

Next, we set up a system of equations for the components of the vector equation: \( \begin{cases} ae^{t} + be^{t} = 0 \\ -ae^{t} + be^{t} = 0 \end{cases} \)

Step 4: Solve the system of equations

Now we can find values of a and b that satisfy the above system of equations. The first equation can be written as: $$ (a+b)e^t =0 $$ Since e^t can never be zero, therefore (a+b) must be zero. The second equation can also be written as: $$ (b-a)e^t =0 $$ Again, since e^t can never be zero, therefore (b-a) must be zero. From the two conditions, (a+b) = 0 and (b-a) = 0, it is clear that the only solution for this system is \(a=0\) and \(b=0\).

Step 5: Conclusion

Since the only solution to the system of equations is when \(a=0\) and \(b=0\), it demonstrates that the given vector functions are linearly independent on \((-\infty, \infty)\).

Key Concepts

Vector Functions

Vector functions are functions that map a real number, often representing time or another parameter, to a vector in a multi-dimensional space. In the context of the problem, we have two vector functions,

• \( \mathbf{x}_{1}(t) = \begin{bmatrix} e^t \ -e^t \end{bmatrix} \)
• \( \mathbf{x}_{2}(t) = \begin{bmatrix} e^t \ e^t \end{bmatrix} \)

Here, \( t \) is the parameter, and each function returns a vector with two components. In general terms, vector functions are used to describe trajectories or to express changes in a multi-dimensional space.
In problems involving linear independence, understanding vector functions helps in examining if one vector can be expressed through a combination of others.

System of Equations

A system of equations consists of multiple equations that need to be satisfied at the same time. In our exercise, we derived two equations from the linear combination

• \( ae^{t} + be^{t} = 0 \)
• \( -ae^{t} + be^{t} = 0 \)

These equations describe conditions that values of \( a \) and \( b \) must satisfy so that their linear combination results in the zero vector.
This means both conditions need to be true simultaneously. Solving this system allows us to determine if such values exist, and thus, whether the vector functions are linearly independent. Exploratory steps in these equations often involve algebraic manipulation to find consistent solutions that satisfy all given conditions.

Linear Combination

A linear combination involves expressing a vector as a sum of scalar multiples of other vectors. This is written as \[ a\mathbf{x}_{1}(t) + b\mathbf{x}_{2}(t) = \mathbf{0} \]where \(a\) and \(b\) are scalars, and the goal is to check if the only solution where this sum equals the zero vector is the trivial solution: \(a = 0\) and \(b = 0\).

Linear combinations are crucial because they help us explore the span of vector functions - whether they can cover the entire space or just a subset.
In scenarios where only the trivial solution is possible, it demonstrates the vectors are independent, having no shared 'directions' or overlap in space.

Scalars

Scalars are single numbers that scale a vector without altering its direction in space. In the equation \( a\mathbf{x}_{1}(t) + b\mathbf{x}_{2}(t) = \mathbf{0} \), \(a\) and \(b\) are the scalars in question. They represent the weights or coefficients given to each vector function.

• If the scalars result in the linear combination forming the zero vector only with all scalars as zero, the associated vectors are linearly independent.
• Scalars, in this context, facilitate finding solutions that tell us about relationships among vector functions in terms of dependence or independence.

Understanding scalars is fundamental because they modify the scale of vectors, which in turn impacts the solutions to the system of equations formed by the linear combination.