Using vector addition and scaling together, we can generate new vectors from old ones. Suppose [math]v_1,v_2,\ldots,v_p[/math] be p vectors in [math]\mathbb{R}^n[/math] (where n can be 2, 3 and any other natural number). Let [math]c_1,c_2,\ldots,c_p[/math] be p real numbers. Then we can define a new vector v as follows:[br][br][center][math]v=c_1v_1+c_2v_2+\cdots+c_pv_p[/math][br][/center][br]This new vector v is called a [b]linear combination[/b] of [math]v_1,v_2,\ldots,v_p[/math] with [b]weights[/b] [math]c_1,c_2,\ldots,c_p[/math].[br][br]In the applet below, you can see the linear combination of vectors u and v in [math]\mathbb{R}^2[/math] with integer weights (range from -10 to 10). Here we regard vectors as points instead of arrows for better illustration. In the input bar, you can type any such linear combination (e.g. 3u-v) and the arrow pointing to the corresponding point will appear.[br][br]
Given any vectors [math] v_1, v_2, \ldots, v_p [/math] in [math]\mathbb{R}^n[/math], we consider the set of all linear combinations of them. It is called the set of vectors in [math]\mathbb{R}^n[/math] [b]spanned (or generated)[/b] by [math] v_1, v_2, \ldots, v_p [/math]. This set is denoted by [math]\text{Span} \{v_1, v_2, \ldots, v_p \}[/math]. In other words, we have[br][br][math]\text{Span} \{v_1, v_2, \ldots, v_p \}=\{c_1v_1+c_2v_2+\cdots+c_pv_p\in \mathbb{R}^n| \ c_1, c_2, \ldots, c_p\in \mathbb{R}\}[/math][br][br]In the applet, you can study [math]\text{Span}\{u,v\}[/math] for different u and v. Then answer the following questions:
What can you say about u and v when [math] \text{Span}\{u,v\}=\mathbb{R}^2[/math] ?
u is not a scalar multiple of v and v is not a scalar multiple of u.
Find out all the possibilities for [math] \text{Span}\{u,v\}[/math].
There are three possibilities:[br][list=1][*]When u and v are both zero vectors, their span is the set containing only the zero vector.[/*][*]When either u or v is a non-zero vector and one is a scalar multiple of another, their span is a line through the origin that contains the non-zero vector.[/*][*]When u and v are non-zero vector and one is not a scalar multiple of another, their span is [math]\mathbb{R}^2[/math].[br][/*][/list]
Given three vectors u, v and w in [math]\mathbb{R}^3[/math], we will study the following subsets: [math] \text{Span}\{u\}, \text{Span}\{u,v\}[/math] and [math] \text{Span}\{u,v,w\}[/math]. You can drag the three vectors freely in the applet below.
Given any vectors [math]v_1, v_2, \ldots, v_p [/math] in [math] \mathbb{R}^3 [/math]. Find out all the possibilities of [math]\text{Span}\{v_1, v_2, \ldots, v_p\}[/math].
There are four possibilities:[br][list=1][*]If all vectors in the set are zero vectors, then the span of the set is the set containing only the zero vector.[/*][*]If at least one vector in the set is non-zero and all other vectors are a scalar multiple of this non-zero vector, then the span of the set is the line through the origin that contains this non-zero vector.[/*][*]If at least two vectors in the set are non-zero so that the span of these two vectors is a plane through the origin, and each remaining vector is a linear combination of these two vectors, then the span of the set is that plane.[/*][*]If at least three vectors in the set are non-zero so that among them no one vector is a linear combination of the remaining two vectors, then the span of the set is [math]\mathbb{R}^{^3}[/math].[br][/*][/list][*] [br][/*]
Now we express u, v, and w In [math] \mathbb{R}^3 [/math] as column vectors:[br][br][math] u=\begin{pmatrix} u_x \\ u_y \\ u_z\end{pmatrix}, v= \begin{pmatrix} v_x \\ v_y \\ v_z\end{pmatrix}, w= \begin{pmatrix} w_x \\ w_y \\ w_z\end{pmatrix}[/math][br][br]We can write the linear combination of them with weights [math]c_1, c_2, c_3 [/math] as follows:[br][br][math]c_1u+c_2v+c_3w=c_1 \begin{pmatrix} u_x \\ u_y \\ u_z\end{pmatrix}+c_2 \begin{pmatrix} v_x \\ v_y \\ v_z\end{pmatrix}+c_3 \begin{pmatrix} w_x \\ w_y \\ w_z\end{pmatrix}= \begin{pmatrix} c_1u_x+c_2v_x+c_3w_x \\ c_1u_y+c_2v_y+c_3w_y \\ c_1u_z+c_2v_z+c_3w_z\end{pmatrix}[/math][br][br]For vectors in another dimensions, you can get a similar column vector for their linear combination. Observe that the arithmetic on column vectors is essentially the arithmetic on the corresponding “entries” on the same row.[br][br]Such matrix representation of linear combination is closely related to [b]systems of linear equations[/b], which we will study in detail later.[br][br]