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Show that the functions $\{1 + 2t, 3 − 2t, −1 + 7t\}$ are linearly dependent by writing one of these functions as a linear combination of the other two

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Would you be more comfortable writing one of the vectors $(1,2)$, $(3,-2)$, and $(-1,7)$ as a linear combination of the other two? It is effectively the same problem. – André Nicolas Mar 29 '12 at 13:49
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Just a sidenote (and maybe not relevant for you): When saying that something are linearly dependent, you have to say "over what". In this case, your functions are linearly dependent over $\mathbb{R}$, the real numbers. – Fredrik Meyer Mar 29 '12 at 18:30

2 Answers 2

Choose one of the three functions, say $-1 + 7t$. If it can be written as a linear combination of the other two, then there exist constants $a$ and $b$ such that $$-1+7t = a(1+2t) + b(3-2t)$$

This has to hold for all values of $t$, so the coefficient of $t$ on the left must match that on the right; similarly, the coefficient of the units must match on both sides. This will give you a system of equations for $a$ and $b$.

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The following is good if you have more functions.

We put these functions into a matrix, $M$ and do row and column manipulations as follow:

$$M=\begin{pmatrix} 1 &2 \\ 3 &-2 \\ -1 & 7 \end{pmatrix}$$

$$\sim\begin{pmatrix} 1 &2 \\ 3 &-2 \\ 0 & 9 \end{pmatrix}$$

$$\sim\begin{pmatrix} 1 &2 \\ 3 &-2 \\ 0 & 1 \end{pmatrix}$$

$$\sim\begin{pmatrix} 1 &0 \\ 3 &-2 \\ 0 & 1 \end{pmatrix}$$

We can see that, $R_2$ is linear combinations of $R_1$ and $R_3$, $R_2=3R_1-2R_3$

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