Matrix multiplication allows us to rewrite a linear system in the form \(A\xvec = \bvec\text<.>\) Besides being a more compact way of expressing a linear system, this form allows us to think about linear systems geometrically since matrix multiplication is defined in terms of linear combinations of vectors.
We now return to our two fundamental questions, rephrased here in terms of matrix multiplication. Existence: Is there a solution to the equation \(A\xvec=\bvec\text\) Uniqueness: If there is a solution to the equation \(A\xvec=\bvec\text\) is it unique?In this section, we focus on the existence question and see how it leads to the concept of the span of a set of vectors.
If the equation \(A\xvec = \bvec\) is inconsistent, what can we say about the pivot positions of the augmented matrix \(\left[\begin
\begin A = \left[ \begin 1 \amp 0 \amp -2 \\ -2 \amp 2 \amp 2 \\ 1 \amp 1 \amp -3 \end\right]\text <.>\end
If \(\bvec=\threevec<2><2>\text\) is the equation \(A\xvec = \bvec\) consistent? If so, find a solution. If \(\bvec=\threevec<2><2>\text\) is the equation \(A\xvec = \bvec\) consistent? If so, find a solution. Identify the pivot positions of \(A\text<.>\)For our two choices of the vector \(\bvec\text<,>\) one equation \(A\xvec = \bvec\) has a solution and the other does not. What feature of the pivot positions of the matrix \(A\) tells us to expect this?
In the preview activity, we considered a \(3\times3\) matrix \(A\) and found that the equation \(A\xvec = \bvec\) has a solution for some vectors \(\bvec\) in \(\real^3\) and has no solution for others. We will introduce a concept called span that describes the vectors \(\bvec\) for which there is a solution.
We can write an \(m\times n\) matrix \(A\) in terms of its columns \begin A = \left[\begin \vvec_1\amp\vvec_2\amp\cdots\amp\vvec_n \end\right]\text \endRemember that Proposition 2.2.4 says that the equation \(A\xvec = \bvec\) is consistent if and only if we can express \(\bvec\) as a linear combination of \(\vvec_1,\vvec_2,\ldots,\vvec_n\text<.>\)
The of a set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) is the set of all linear combinations that can be formed from the vectors.
Alternatively, if \(A = \begin
is one vector in the span of the vectors \(\vvec\) and \(\wvec\) because it is a linear combination of \(\vvec\) and \(\wvec\text<.>\)
To determine whether the vector \(\bvec=\twovec<5>\) is in the span of \(\vvec\) and \(\wvec\text\) we form the matrix
\begin A = \begin \vvec \amp \wvec \end = \begin -2 \amp 8 \\ 1 \amp -4 \\ \end \end and consider the equation \(A\xvec=\bvec\text<.>\) We have\begin \left[ \begin
which shows that the equation \(A\xvec = \bvec\) is inconsistent. Therefore, \(\bvec=\twovec52\) is one vector that is not in the span of \(\vvec\) and \(\wvec\text<.>\)
The figure shows us that \(\bvec = \vvec + 2\wvec = \twovec02\) is a linear combination of \(\vvec\) and \(\wvec\text<.>\) Indeed, we can verify this algebraically by constructing the linear system
\begin \begin\vvec \amp \wvec \end ~ \xvec = \twovec02, \end whose corresponding augmented matrix has the reduced row echelon form\begin \left[ \begin
Because this system is consistent, we know that \(\bvec=\twovec02\) is in the span of \(\vvec\) and \(\wvec\text<.>\)
In fact, we can say more. Notice that the coefficient matrix \begin \begin 2 \amp -1 \\ 0 \amp 1 \\ \end \sim \begin 1 \amp 0 \\ 0 \amp 1 \\ \end \endhas a pivot position in every row. This means that for any other vector \(\bvec\text<,>\) the augmented matrix corresponding to the equation \(\begin\vvec \amp \wvec \end ~\xvec = \bvec\) cannot have a pivot position in its rightmost column:
\begin \left[ \begin
Therefore, the equation \(\begin
The intuitive meaning is that we can walk to any point in the plane by moving an appropriate distance in the \(\vvec\) and \(\wvec\) directions.
From the figure, we expect that \(\bvec = \twovec02\) is not a linear combination of \(\vvec\) and \(\wvec\text<.>\) Once again, we can verify this algebraically by constructing the linear system
\begin \begin\vvec \amp \wvec \end ~ \xvec = \twovec02. \end The augmented matrix has the reduced row echelon form\begin \left[ \begin
from which we see that the system is inconsistent. Therefore, \(\bvec=\twovec02\) is not in the span of \(\vvec\) and \(\wvec\text<.>\)
We should expect this behavior from the coefficient matrix \begin \begin -1 \amp 2 \\ 1 \amp -2 \\ \end \sim \begin 1 \amp -2 \\ 0 \amp 0 \\ \end. \endBecause the second row of the coefficient matrix does not have a pivot position, it is possible for a linear system \(\begin
\begin \left[ \begin
If we notice that \(\wvec = -2\vvec\text<,>\) we see that any linear combination of \(\vvec\) and \(\wvec\text<,>\)
\begin c\vvec + d\wvec = c\vvec -2d\vvec = (c-2d)\vvec, \endis actually a scalar multiple of \(\vvec\text<.>\) Therefore, the span of \(\vvec\) and \(\wvec\) is the line defined by the vector \(\vvec\text<.>\) Intuitively, this means that we can only walk to points on this line using these two vectors.
In Example 2.3.5, we saw that \(\laspan = \real^2\text<.>\) However, for the vectors in Example 2.3.7, we saw that \(\laspan\) is simply a line.
A set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) naturally defines a matrix \(A = \begin
The previous examples point to the fact that the span is related to the pivot positions of \(A\text<.>\) While Section 2.4 and Section 3.5 develop this idea more fully, we will now examine the possibilities in \(\real^3\text<.>\)
Suppose \(\vvec=\threevec<1><1>\text<.>\) Give a geometric description of \(\laspan\) and a rough sketch of \(\vvec\) and its span in Figure 2.3.10.
Now consider the two vectors \begin \evec_1 = \threevec,~~~ \evec_2 = \threevec\text \endSketch the vectors below. Then give a geometric description of \(\laspan\) and a rough sketch of the span in Figure 2.3.11.
Let’s now look at this situation algebraically by writing write \(\bvec = \threevec
Form the matrix \(\left[\begin
If the span of a set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) is \(\real^3\text<,>\) what can you say about the pivot positions of the matrix \(\left[\begin \vvec_1\amp\vvec_2\amp\ldots\amp\vvec_n \end\right]\text\)
What is the smallest number of vectors such that \(\laspan <\vvec_1,\vvec_2,\ldots,\vvec_n>= \real^3\text\) The types of sets that appear as the span of a set of vectors in \(\real^3\) are relatively simple.First, with a single nonzero vector, all linear combinations are simply scalar multiples of that vector so that the span of this vector is a line, as shown in Figure 2.3.12.
Notice that the matrix formed by this vector has one pivot position. For example, \begin \threevec \sim \threevec\text \endThe span of two vectors in \(\real^3\) that do not lie on the same line will be a plane, as seen in Figure 2.3.13.
For example, the vectors \begin \vvec_1=\threevec,~~~ \vvec_2=\threevec \end lead to the matrix\begin \left[\begin -2 \amp 1 \\ 3 \amp -1 \\ 1 \amp 3 \\ \end\right] \sim \left[\begin 1 \amp 0 \\ 0 \amp 1 \\ 0 \amp 0 \\ \end\right] \end
with two pivot positions. Finally, a set of three vectors, such as \begin \vvec_1=\threevec12,~~~ \vvec_2=\threevec201,~~~ \vvec_3=\threevec20 \end may form a matrix having three pivot positions\begin \left[\begin \vvec_1 \amp \vvec_2 \amp \vvec_3 \end\right] = \left[\begin 1 \amp 2 \amp -2 \\ 2 \amp 0 \amp 2 \\ -1 \amp 1 \amp 0 \\ \end\right] \sim \left[\begin 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end\right], \end
one in every row. When this happens, no matter how we augment this matrix, it is impossible to obtain a pivot position in the rightmost column:
\begin \left[\begin
Therefore, any linear system \(\begin
To summarize, we looked at the pivot positions in a matrix whose columns are the three-dimensional vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\text<.>\) We found that with
one pivot position, the span was a line. two pivot positions, the span was a plane. three pivot positions, the span was \(\real^3\text<.>\)Though we will return to these ideas later, for now take note of the fact that the span of a set of vectors in \(\real^3\) is a relatively simple, familiar geometric object.
The reasoning that led us to conclude that the span of a set of vectors is \(\real^3\) when the associated matrix has a pivot position in every row applies more generally.
Suppose we have vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) in \(\real^m\text<.>\) Then \(\laspan<\vvec_1,\vvec_2,\ldots,\vvec_n>=\real^m\) if and only if the matrix \(\left[\begin \vvec_1\amp\vvec_2\amp\cdots\amp\vvec_n \end\right]\) has a pivot position in every row.
This tells us something important about the number of vectors needed to span \(\real^m\text<.>\) Suppose we have \(n\) vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) that span \(\real^m\text<.>\) The proposition tells us that the matrix \(A = \left[\begin \vvec_1\amp\vvec_2\amp\ldots\amp\vvec_n \end\right]\) has a pivot position in every row, such as in this reduced row echelon matrix.
\begin \left[\begin 1 \amp 0 \amp * \amp 0 \amp * \amp 0 \\ 0 \amp 1 \amp * \amp 0 \amp * \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \amp * \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 1 \\ \end\right]. \end
Since a matrix can have at most one pivot position in a column, there must be at least as many columns as there are rows, which implies that \(n\geq m\text<.>\) For instance, if we have a set of vectors that span \(\real^\text\) there must be at least 632 vectors in the set.
We have thought about a linear combination of a set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) as the result of walking a certain distance in the direction of \(\vvec_1\text<,>\) followed by walking a certain distance in the direction of \(\vvec_2\text<,>\) and so on. If \(\laspan <\vvec_1,\vvec_2,\ldots,\vvec_n>= \real^m\text<,>\) this means that we can walk to every point in \(\real^m\) using the directions \(\vvec_1,\vvec_2,\ldots,\vvec_n\text<.>\) Intuitively, this proposition is telling us that we need at least \(m\) directions to have the flexibility needed to reach every point in \(\real^m\text<.>\)
Because span is a concept that is connected to a set of vectors, we say, “The span of the set of vectors \(\vvec_1, \vvec_2, \ldots, \vvec_n\) is . ” While it may be tempting to say, “The span of the matrix \(A\) is . ” we should instead say “The span of the columns of the matrix \(A\) is . ”
We defined the span of a set of vectors and developed some intuition for this concept through a series of examples.
The span of a set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) is the set of linear combinations of the vectors. We denote the span by \(\laspan<\vvec_1,\vvec_2,\ldots,\vvec_n>\text<.>\)