Determine whether the following vector-valued functions are...

Determine whether the following vector-valued functions are linearly dependent or linearly independent: (a) $\left(\begin{array}{l}1 \\ t\end{array}\right),\left(\begin{array}{r}-t \\ 1\end{array}\right)$, (b) $\left(\begin{array}{c}1+t \\ t\end{array}\right),\left(\begin{array}{c}1-t^2 \\ t-t^2\end{array}\right)$ (c) $\left(\begin{array}{l}1 \\ t\end{array}\right),\left(\begin{array}{l}t \\ 2\end{array}\right),\left(\begin{array}{r}-t \\ t\end{array}\right)$, (d) $\left(\begin{array}{c}e^{-t} \\ -e^t\end{array}\right),\left(\begin{array}{c}-e^{-t} \\ e^t\end{array}\right)$ (e) $\left(\begin{array}{r}e^{2 t} \cos 3 t \\ -e^{2 t} \sin 3 t\end{array}\right),\left(\begin{array}{c}e^{2 t} \sin 3 t \\ e^{2 t} \cos 3 t\end{array}\right)$, (f) $\left(\begin{array}{c}\cos 3 t \\ \sin 3 t\end{array}\right),\left(\begin{array}{c}\sin 3 t \\ \cos 3 t\end{array}\right)$, (g) $\left(\begin{array}{c}1 \\ t \\ 1-t\end{array}\right),\left(\begin{array}{c}0 \\ -2 \\ 2\end{array}\right),\left(\begin{array}{c}3 \\ 1+3 t \\ 2-3 t\end{array}\right)$. (h) $\left(\begin{array}{r}e^t \\ -e^t \\ e^t\end{array}\right),\left(\begin{array}{r}e^t \\ e^t \\ -e^t\end{array}\right)+\left(\begin{array}{r}-e^t \\ e^t \\ e^t\end{array}\right)$, (i) $\left(\begin{array}{c}e^t \\ t e^t \\ t^2 e^t\end{array}\right),\left(\begin{array}{c}t^2 e^t \\ e^t \\ t e^t\end{array}\right),\left(\begin{array}{c}t e^t \\ t^2 e^t \\ e^t\end{array}\right),\left(\begin{array}{c}e^t \\ e^t \\ e^t\end{array}\right)$.

Determine whether the following vector-valued functions are linearly dependent or linearly independent:
(a) $\left(\begin{array}{l}1 \\ t\end{array}\right),\left(\begin{array}{r}-t \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{c}1+t \\ t\end{array}\right),\left(\begin{array}{c}1-t^2 \\ t-t^2\end{array}\right)$
(c) $\left(\begin{array}{l}1 \\ t\end{array}\right),\left(\begin{array}{l}t \\ 2\end{array}\right),\left(\begin{array}{r}-t \\ t\end{array}\right)$,
(d)
$\left(\begin{array}{c}e^{-t} \\ -e^t\end{array}\right),\left(\begin{array}{c}-e^{-t} \\ e^t\end{array}\right)$
(e) $\left(\begin{array}{r}e^{2 t} \cos 3 t \\ -e^{2 t} \sin 3 t\end{array}\right),\left(\begin{array}{c}e^{2 t} \sin 3 t \\ e^{2 t} \cos 3 t\end{array}\right)$,
(f) $\left(\begin{array}{c}\cos 3 t \\ \sin 3 t\end{array}\right),\left(\begin{array}{c}\sin 3 t \\ \cos 3 t\end{array}\right)$,
(g) $\left(\begin{array}{c}1 \\ t \\ 1-t\end{array}\right),\left(\begin{array}{c}0 \\ -2 \\ 2\end{array}\right),\left(\begin{array}{c}3 \\ 1+3 t \\ 2-3 t\end{array}\right)$.
(h) $\left(\begin{array}{r}e^t \\ -e^t \\ e^t\end{array}\right),\left(\begin{array}{r}e^t \\ e^t \\ -e^t\end{array}\right)+\left(\begin{array}{r}-e^t \\ e^t \\ e^t\end{array}\right)$,
(i) $\left(\begin{array}{c}e^t \\ t e^t \\ t^2 e^t\end{array}\right),\left(\begin{array}{c}t^2 e^t \\ e^t \\ t e^t\end{array}\right),\left(\begin{array}{c}t e^t \\ t^2 e^t \\ e^t\end{array}\right),\left(\begin{array}{c}e^t \\ e^t \\ e^t\end{array}\right)$.

Key Concepts

Linear Combination

A linear combination involves adding together scalar multiples of functions. In the context of vector-valued functions, this means multiplying each function by a constant and summing the results to form a new function. Determining if a set is linearly independent or dependent hinges on whether nontrivial (non-zero) combinations exist that yield the zero vector function for all values of the parameter.

Vector-Valued Functions

These functions map a variable, typically time or another parameter, into vectors in a finite-dimensional space. Each function is represented by multiple component functions, and operations such as addition and scalar multiplication are performed component-wise. Understanding their structure is essential when discussing concepts like linear independence in a vector space context.

Coefficient Comparison

When checking for linear independence of vector-valued functions, one typically sets up an equation where a linear combination of the functions equals the zero vector for every parameter value. By equating corresponding components, this generates a system of equations in terms of the constant coefficients. Solving this system reveals whether the only solution is the trivial one (all coefficients zero) or if nontrivial solutions exist, thus determining independence or dependence.

Linear Independence

A set of vector-valued functions is said to be linearly independent if the only constant coefficients that can form a linear combination of these functions to yield the zero vector function (for every value of the independent variable) are all zero. This concept is vital because it helps determine whether the functions provide unique directions in the function space and are not redundant in representing solutions or states.

Linear Dependence

Conversely, a set of vector-valued functions is linearly dependent if there exist some nonzero constants (not all zero) such that a linear combination of these functions gives the zero vector for every value of the variable. This implies that at least one of the functions can be expressed as a combination of the others, highlighting redundancy within the set.

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