It is a remarkable fact that algebra, which is about symbolic equations and their solutions, and geometry are intimately connected. For instance, the solution set of a linear equation in two unknowns, such as , can be represented graphically by a straight line. The aim of this section is to further this connection by introducing vectors, which will help us to apply geometric intuition to our thinking about linear systems.
A vector is most simply thought of as a matrix with a single column. For instance, and are both vectors. The entries in a vector are called its components. Since the vector has two components, we say that it is a two-dimensional vector; in the same way, the vector is a four-dimensional vector.
While it can be difficult to visualize a four-dimensional vector, we can draw a simple picture describing the two-dimensional vector , as shown in Figure 2.1.1.
Figure2.1.1.A graphical representation of the vector .
We can think of as describing a walk in the plane where we move two units horizontally and one unit vertically. Though we allow ourselves to begin walking from any point in the plane, we will most frequently begin at the origin in which case we arrive at the the point , as shown in the figure.
Consider vectors that have the form where is any scalar. Sketch a few of these vectors when, say, and . Give a geometric description of this set of vectors.
First, we see that scalar multiplication has the effect of stretching or compressing a vector. Multiplying by a negative scalar changes the direction of the vector. In either case, Figure 2.1.5 shows that a scalar multiple of a vector lies on the same line defined by .
To represent the sum , we imagine walking from the origin with the appropriate horizontal and vertical changes given by . From there, we continue our walk using the horizontal and vertical changes prescribed by , after which we arrive at the sum . This is illustrated on the left of Figure 2.1.6 where the tail of is placed on the tip of .
Figure2.1.6.Vector addition as a simple walk in the plane is illustrated on the left. The vector sum is represented as the diagonal of a parallelogram on the right.
Alternatively, we may construct the parallelogram with and as two sides. The sum is then the diagonal of the parallelogram, as illustrated on the right of Figure 2.1.6.
We have now seen that the set of vectors having the form is a line. To form the set of vectors , we can begin with the vector and add multiples of . Geometrically, this means that we begin from the tip of and move in a direction parallel to . The effect is to translate the line by the vector , as shown in Figure 2.1.7.
At times, it will be useful for us to think of vectors and points interchangeably. That is, we may wish to think of the vector as describing the point and vice-versa. When we say that the vectors having the form form a line, we really mean that the tips of the vectors all lie on the line passing through and parallel to .
Sage can perform scalar multiplication and vector addition. We define a vector using the vector command; then * and + denote scalar multiplication and vector addition.
Linear combinations, which we encountered in the preview activity, provide the link between vectors and linear systems. In particular, they will help us apply geometric intuition to problems involving linear systems.
The diagram below can be used to construct linear combinations whose weights and may be varied using the sliders at the top. The vectors and are outlined while the linear combination
What is the linear combination of and when and ? You may find this result using the diagram, but you should also verify it by computing the linear combination.
This activity illustrates how linear combinations are constructed geometrically: the linear combination is found by walking along a total of times followed by walking along a total of times. When one of the weights is held constant while the other varies, the vector moves along a line.
The previous activity also shows that questions about linear combinations lead naturally to linear systems. Suppose we have vectors and . Let’s determine whether we can describe the vector as a linear combination of and . In other words, we would like to know whether there are weights and such that
To solve this linear system, we construct its corresponding augmented matrix and find its reduced row echelon form,
giving us the weights and ; that is,
.
In fact, we know more because the reduced row echelon matrix tells us that these are the only possible weights. Therefore, may be expressed as a linear combination of and in exactly one way.
This example demonstrates the connection between linear combinations and linear systems. Asking whether a vector is a linear combination of vectors is equivalent to asking whether an associated linear system is consistent.
In fact, we may easily describe the associated linear system in terms of the vectors ,, and . Notice that the augmented matrix we found in our example was The first two columns of this matrix are and and the rightmost column is . As shorthand, we will write this augmented matrix replacing the columns with their vector representation:
.
This fact is generally true so we record it in the following proposition.
Activity2.1.3.Linear combinations and linear systems.
Given the vectors
,
can be expressed as a linear combination of ,, and ? Rephrase this question by writing a linear system for the weights ,, and and use the Sage cell below to answer this question.
Identify vectors ,,, and such that the question "Is this linear system consistent?" is equivalent to the question "Can be expressed as a linear combination of ,, and ?"
Considering the vectors ,, and from the previous part, can we write every three-dimensional vector as a linear combination of these vectors? Explain how the pivot positions of the matrix help answer this question.
Considering the vectors ,, and from the previous part, can we write every three-dimensional vector as a linear combination of these vectors? Explain how the pivot positions of the matrix help answer this question.
These vectors appear to lie on the same line, a fact that becomes apparent once we notice that . Intuitively, we think of the linear combination
as the result of walking times in the direction and times in the direction. With these vectors, we are always walking along the same line so it would seem that any linear combination of these vectors should lie on the same line. In addition, a vector that is not on the line, say , should be not be expressible as a linear combination of and .
Notice that the reduced row echelon form of the coefficient matrix
tells us to expect this. Since there is not a pivot position in the second row of the coefficient matrix , it is possible for a pivot position to appear in the rightmost column of the augmented matrix
Nutritional information about a breakfast cereal is printed on the box. For instance, one serving of Frosted Flakes has 111 calories, 140 milligrams of sodium, and 1.2 grams of protein. We may represent this as a vector
.
One serving of Cocoa Puffs has 120 calories, 105 milligrams of sodium, and 1.0 grams of protein.
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Write the vector describing the nutritional content of Cocoa Puffs.
Suppose you eat servings of Frosted Flakes and servings of Cocoa Puffs. Use the language of vectors and linear combinations to express the quantities of calories, sodium, and protein you have consumed.
Suppose is a collection of -dimensional vectors and that the matrix has a pivot position in every row. If is any -dimensional vector, then can be written as a linear combination of .
Suppose is a collection of -dimensional vectors and that the matrix has a pivot position in every row and every column. If is any -dimensional vector, then can be written as a linear combination of in exactly one way.
A theme that will later unfold concerns the use of coordinate systems. We can identify the point with the tip of the vector , drawn emanating from the origin. We can then think of the usual Cartesian coordinate system in terms of linear combinations of the vectors
For instance, the point is identified with the vector
Figure2.1.15.The usual Cartesian coordinate system, defined by the vectors and , is shown on the left along with the representation of the point . The right shows a nonstandard coordinate system defined by vectors and .
What is the general strategy for converting a point from standard Cartesian coordinates to the new coordinates ? Actually implementing this strategy in general may take a bit of work so just describe the strategy. We will study this in more detail later.