Lecture 7 - A Start into Chapter 2

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Starting Chapter 2

Suppose ${\vec{v}}_{1}, . . ., {\vec{v}}_{n} \in V$ . If we scale each one of them:

α_{1} {\vec{v}}_{1}, . . ., α_{n} {\vec{v}}_{n} \in V

And:

α_{1} {\vec{v}}_{1} + . . . + α_{n} {\vec{v}}_{n} \in V

Because both of closure under vector addition and scalar multiplication. We give a special name to this, calling it a linear combination:

Linear Combination

Given vectors ${\vec{v}}_{1}, . . ., {\vec{v}}_{n} \in V$ , a linear combination of these vectors is a vector of the form:

α_{1} {\vec{v}}_{1} + . . . + α_{n} {\vec{v}}_{n} \in V

Where each $α_{i} \in F$ .

Note that subspaces are thus closed under linear combinations.

Let's look at some examples:

Example

Is $(7, 4, 0)$ a linear combination of ${\vec{v}}_{1} = (0, 1, 0)$ , ${\vec{v}}_{2} = (1, - 1, 0)$ ?

\begin{proof}
Notice that:

(7, 4, 0) = 11 (0, 1, 0) + 7 (1, - 1, 0) = 11 {\vec{v}}_{1} + 7 {\vec{v}}_{2}

So yes it's a linear combination.
\end{proof}

Example

Is the function $\sqrt{2} \sin (x + \frac{π}{4})$ a linear combination of $\sin (x)$ and $\cos (x)$ ?

\begin{proof}
It is a linear combination since, using $\sin (α + β) = \sin (α) \cos (β) + \cos (α) \sin (β)$ :

\sqrt{2} \sin (x + \frac{π}{4}) = \sqrt{2} (\sin (x) \cos (π / 4) + \cos (x) \sin (π / 4)) = \sin (x) + \cos (x)

So yes it is a linear combination. (you can note that in this case $V = R^{R}$ ).
\end{proof}

Example

Is $e^{x}$ a L.C. of $\sin (x)$ and $\cos (x)$ ?

\begin{proof}
The answer is no! Notice that any linear combination of $\sin, \cos$ must live within some rance $[a, b]$ for any $a, b \in R$ . But $e^{x} \in [0, \infty]$ which breaks this pattern, so it can't be a linear combination of the two vectors in question.

Also, we can show that vectors in our subspace are periodic, while $e^{x}$ definitely is not.
\end{proof}