Standard Cartesian coordinates are commonly used to describe points in the plane. If we mention the point , we know that we arrive at this point from the origin by moving four units to the right and three units up.
Sometimes, however, it is more natural to work in a different coordinate system. Suppose that you live in the city whose map is shown in Figure 3.2.1 and that you would like to give a guest directions for getting from your house to the store. You would probably say something like, "Go four blocks up Maple. Then turn left on Main for three blocks." The grid of streets in the city gives a more natural coordinate system than standard north-south, east-west coordinates.
In this section, we will develop the concept of a basis through which we will create new coordinate systems in . We will see that the right choice of a coordinate system provides a more natural way to approach some problems.
In the preview activity, we worked with a set of two vectors in and found that we could express any vector in in two different ways: in the usual way where the components of the vector describe horizontal and vertical changes, and in a new way as a linear combination of and . We could also translate between these two descriptions. This example illustrates the central idea of this section.
In the preview activity, we created a new coordinate system for using linear combinations of a set of two vectors. More generally, the following definition will guide us.
We know that the span of the set of vectors is if and only if has a pivot position in every row. We also know that the set of vectors is linearly independent if and only if has a pivot position in every column. This means that a set of vectors forms a basis if and only if has a pivot in every row and every column. Therefore, must be row equivalent to the identity matrix :
In addition to helping identify bases, this fact tells us something important about the number of vectors in a basis. Since the matrix has a pivot position in every row and every column, it must have the same number of rows as columns. Therefore, the number of vectors in a basis for must be . For example, a basis for must have exactly 23 vectors.
form the columns of the identity matrix, which implies that this set forms a basis for . More generally, the set of vectors forms a basis for , which we call the standard basis for .
A basis for forms a coordinate system for , as we will describe. Rather than continuing to write a list of vectors, we will find it convenient to denote a basis using a single symbol, such as
In the standard coordinate system, the point is found by moving 2 units to the right and 3 units down. We would like to define a new coordinate system where we interpret to mean we move two times along and 3 times along . As we see in the figure, doing so leaves us at the point , expressed in the usual coordinate system.
Since we now have two descriptions of the vector , we need some notation to keep track of which coordinate system we are using. Because , we will write
.
More generally, will denote the coordinates of in the basis ; that is, is the vector of weights such that
Suppose we know the expression of a vector in standard coordinates. How can we find its coordinates in the basis ? For instance, suppose and that we would like to find . We can write
which means that
or
This linear system for the weights defines an augmented matrix
This example illustrates how a basis in provides a new coordinate system for and shows how we may translate between this coordinate system and the standard one.
More generally, suppose that is a basis for . We know that the span of the vectors is , which implies that any vector in can be written as a linear combination of the vectors. In addition, we know that the vectors are linearly independent, which means that we can write as a linear combination of the vectors in exactly one way. Therefore, we have
where the weights are unique. In this case, we write the coordinate description of in the basis as
Using what you found in the previous part, find a matrix such that, for any vector , we have . What is the relationship between the two matrices and ? Explain why this relationship holds.
This activity demonstrates how we can efficiently convert between coordinate systems defined by different bases. Letās consider a basis and a vector . We know that
.
If we use to denote the matrix whose columns are the basis vectors, then we find that
where . This means that the matrix converts coordinates in the basis into standard coordinates.
It is relatively straightforward to convert a vectorās representation in this basis into to the standard basis using the matrix whose columns are the basis vectors:
For example, suppose that the vector is described in the coordinate system defined by the basis as . We then have
The average revenue for the first two quarters is 11.7, which is 1.925 million dollars above the yearly average. Similarly, the average revenue for the last two quarters is 1.925 million dollars below the yearly average. This is recorded by the second term
Finally, the first quarterās revenue is 1.400 million dollars below the average over the first two quarters and the second quarterās revenue is 1.400 million dollars above that average. This, and the corresponding data for the last two quarters, is captured by the last two terms:
If we write , we see that the coefficient measures the average revenue over the year, measures the deviation from the annual average in the first and second halves of the year, and measures how the revenue in the first and second quarter differs from the average in the first half of the year. In this way, the coefficients provide a view of the revenue over different time scales, from an annual summary to a finer view of quarterly behavior.
This basis is sometimes called a Haar wavelet basis, and the change of basis is known as a Haar wavelet transform. In the next section, we will see how this basis provides a useful way to store digital images.
An important problem in the field of computer vision is to detect edges in a digital photograph, as is shown in Figure 3.2.12. Edge detection algorithms are useful when, say, we want a robot to locate an object in its field of view. Graphic designers also use these algorithms to create artistic effects.
We will consider a very simple version of an edge detection algorithm to give a sense of how this works. Rather than considering a two-dimensional photograph, we will think about a one-dimensional row of pixels in a photograph. The grayscale values of a pixel measure the brightness of a pixel; a grayscale value of 0 corresponds to black, and a value of 255 corresponds to white.
We can easily see that there is a jump in brightness between pixels 4 and 5, but how can we detect it computationally? We will introduce a new basis for with vectors:
.
Construct the matrix that relates the standard coordinate system with the coordinates in the basis .
Suppose the vectors are expressed in general terms as
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Using the relationship , determine an expression for the coefficient in terms of . What does measure in terms of the grayscale values of the pixels? What does measure in terms of the grayscale values of the pixels?
Readers who are familiar with calculus may recognize that this change of basis converts a vector into , the set of changes in . This process is similar to differentiation in calculus. Similarly, the process of converting into the vector adds together the changes in a process similar to integration. As a result, this change of basis represents a linear algebraic version of the Fundamental Theorem of Calculus.
Suppose we have an invertible matrix , and we perform a sequence of row operations on to form a matrix . Can you guarantee that the columns of form a basis for ?
Suppose you have a set of 10 vectors in and that every vector in can be written as a linear combination of these vectors. Can you guarantee that this set of vectors is a basis for ?
Crystallographers find it convenient to use coordinate systems that are adapted to the specific geometry of a crystal. As a two-dimensional example, consider a layer of graphite in which carbon atoms are arranged in regular hexagons to form the crystalline structure shown in Figure 3.2.14.
The origin of the coordinate system is at the carbon atom labeled by ā0ā. It is convenient to choose the basis defined by the vectors and and the coordinate system it defines.
How do the coordinates of the atoms in the hexagon whose lower left corner is labeled ā1ā compare to the coordinates in the hexagon whose lower left corner is labeled "0"?
You should find that the matrix is a very simple matrix, which means that this basis is well suited to study the effect of multiplication by . This observation is the central idea of the next chapter.