Linear Algebra 340 Hints

Got questions? Ask!. If the answer seems likely to be useful to other students, I will post it here.

Lesson 2

p. 49 #34 Be careful not to confuse the given information here with statement (a) of Theorem 4. Here, we know that the equation Ax = b has a unique solution for one particular vector b. Statement (a) of Theorem 4 says that the equation Ax = b has a solution for all right-hand sides b

Keep in mind the following example:

[ 1  1 ] [ x1 ]      [ 0 ]
[ 1  2 ] [ x2 ]  =   [ 1 ]
[ 1  1 ]             [ 0 ]

This equation has a unique solution, and yet the columns of the matrix do not span R³. Make sure that your response to problem 34 makes explicit use of the difference between that situation and this example.

Lesson 3

Visualizing Linear Transformations.

You need to develop the habit of thinking of a square matrix as as representing a motion of R², R³, etc. The columns of the matrix give the destinations of the standard basis vectors, but there's much more than that: every point in the plane is a linear combination of the standard basis vectors, and linearity means that if the first basis vector gets stretched by a certain factor, then every point has its x coordinate stretched by that factor. Whatever a transformation does to the vector (0, 1), it does the same thing to the y coordinates of all points.

The animated diagram (which requires a JavaScript-enabled browser) illustrates the idea, using the unit circle.

Put your cursor over a matrix to see the effect of multiplying by that matrix.

Lesson 4

p. 132 #11, 12 Notice that the preamble to these problems says all the matrices involved are n x n. This makes a difference to many of the questions! Make sure you carefully consider whether a given statement is true in general, true for square matrices only, or neither. In the second case, exactly how does the squareness of the matrix figure in?

p. 133 #28 Note that we cannot start by writing (AB)^-1 = B^-1A^-1, since we don't know that either A^-1 or B^-1 exists!

Preparing for the Midterm Exam

You have 90 minutes for the exam
You may use a calculator, though it won't be much help. You may not use a Palm Pilot or similar device, or anything with a QWERTY keyboard.
You may not use books or notes.
You do not need to focus on the applications.
You do not need to memorize what Theorem n says.
You DO need to know what theorems are true, but you do not need to cite them by chapter and verse.
You DO need to know the definitions of important terms.
Somewhat less than half the exam is calculation (solve this system of equations...)
A small amount of the exam asks you to actually prove something.
Most of the exam consists of the kind of questions which are meant to test whether you have a good understanding of a topic:
- For which values of a and b is the system consistent?
- What are the connections between the dimensions of the null space and column space?
- Does this set of vectors form a subspace? Why or why not?
- Write the matrix for a given geometric transformation of R².

Lesson 7

p. 223 #20 Note that you don't have to actually prove the needed theorems about continuous functions, merely identify what needs to be proved.

For both (a) and (b), focus on the three properties which characterize a subspace, and show that the sets of functions in (a) and (b) have these properties. In the process, you will need to consider what continuous function corresponds to the zero vector. To do so, consult the definition on page 217.

Lesson 9

Uncovering the Geometric Action of a Matrix: the case A =

This is a fairly nice, symmetric matrix. Let's try to get a feel for the linear transformation T that it represents. The columns tell us where the standard basis vectors e₁ and e₂ get mapped to, and we use that information to draw the image of the unit square. (Place cursor over picture to see animation.)

We can see that both the positive x axis and the positive y axis get mapped into the first quadrant. There is a small amount of stretching horizontally, somewhat more stretching vertically. The general motion within the first quadrant is toward the upper right, somewhat steeper than a 45-degree line.

This description is not bad, but there is a deeper truth to be discovered here. The linear transformation T has certain directions in which it acts. The xy coordinate system does not line up with the characteristic directions of T, so it's hard to see exactly what's going on.

Let's look at the action of T on the unit circle. The circle is isotropic -- has no preferred direction -- so perhaps it will be easier to see the geometric action of T. Put cursor over this diagram to see the linear transformation in action.

Matrix Action, xy Frame of Reference

If we look at this picture with our heads tilted a bit to one side, it looks rather like the action of the diagonal matrix . In fact, let's discard the framework of the x and y axes for the moment, and align our frame of reference with the way T seems to act.

Matrix Action, in its Own Frame of Reference

Look at that! If we simply tilt the picture, and position the direction of greatest stretch on the x axis, we get exactly the action of the diagonal matrix D = .

So the matrix A = is related in some fundamental way to the diagonal matrix D = , whose geometric action is obvious at a glance. How do we identify and use this relationship, and will the same kind of thing hold for other 2 x 2 matrices? What about larger matrices?

The answer lies in the subject of...

Eigenvalues and Eigenvectors

The eigenvalues of A are the numbers 2 and 1, which are exactly the diagonal entries of the related matrix D. These tell us something about the intrinsic action of the linear transformation T: it stretches things by a factor of 2 in some direction, while keeping things unchanged in another direction.
The eigenvectors of A point in the two directions associated with T: the direction in which lengths are streched by a factor of 2, and the direction in which lengths are unchanged. If we write things using a basis of eigenvectors, the matrix will show the action at a glance.
The relationship between A and D is called similarity, and it captures the idea that these two matrices are, deep down, almost identical: both represent the same kind of stretch of R²; the only difference is how each one is aligned with the xy coordinate system. The similarity relationship is just a change of basis, from the standard xy frame of reference to the eigenvector frame of reference.

The great beauty and power of eigenvalues and eigenvectors comes from their alignment with the directions in which a linear transformation acts. This alignment makes the geometric action evident at a glance. It's like finding your way in a city where the roads run from northeast to southwest: if you try to navigate by the standard coordinate system of North, South, East and West, everything is complicated. But in a coordinate system aligned with the directions of the streets, things are much simpler. The eigenvectors give us the proper frame of reference for describing a linear transformation, and the eigenvalues are the simple description in that frame of reference.

Eigen is a German adjective meaning "own" or "innate." The eigenvalues describe the innate action of a linear transformation, independent of how it happens to be aligned with a particular frame of reference. The word "vector" is not German. There are proper English terms: characteristic vector and characteristic value, but they are rarely used; the mixed-language eigenvalue and eigenvector have won out.

Chapters 9-10

David Lay's treatment of most topics is exactly right. One of the very few places where I would take a different approach involves the geometric meaning of eigenvectors and eigenvalues. I'm trying to make up an interactive learning tool about the geometry of eigenvectors and eigenvalues. I've only just started (on 7/26/03), but I figured I'd put the first result up here so people can play with it. Drag the red dots.

This demo requires Macromedia Flash Player 6. If you don't already have it, you can download the free player for Windows or other operating systems.

Lesson 11

The Importance of an Orthogonal Basis. In lessons 9 and 10 we saw that a linear transformation assumed a particularly simple form when expressed with respect to a basis of eigenvectors. In that basis, the separate equations decouple, allowing us to solve each equation on its own.

A similar fact underlies the usefulness of orthogonal (and orthonormal) bases: If we want to write a given vector b as a linear combination of basis vectors {v₁,...v_n}, we usually have to row-reduce an augmented matrix, a process in which all the equations affect each other. But if we have an orthogonal basis {u₁,...u_n}, then we can deal with each basis vector separately, computing the component of b in the direction of each u_i without reference to the other members of the basis. The following exercises show what is happening.

Exercise: Let v₁ = (3, 1) and v₂ = (0, 2). Let b = (5, 5). Compute the projection of b onto each of the v's, and show that the sum of the projections is not b. Draw a sketch of all the relevant vectors.

Important Note: When orthogonality matters, you must use the same scale on x and y axes. Otherwise, orthogonal lines will not appear orthogonal in your sketch.

Exercise: Using the same vectors as above, express b as a linear combination of the v's. Draw a sketch of the relevant vectors.

Understanding Orthogonal Projection. Consider the orthogonal projection of R³ onto the xy plane, defined by T(x, y, z) = (x, y, 0). Algebraically, we " set the third coordinate to zero." Geometrically, we "smash" all of R³ vertically down onto the xy plane.

Exactly the same thing is going on when we project R³ onto any plane W through the origin. If we write coordinates in terms of an orthogonal basis {v₁, v₂, v₃} whose first two members span W, then the orthogonal projection again replaces the third coordinate with zero. And geometrically, the picture is exactly like the projection onto the xy plane; it is merely positioned differently within R³, with the directions of {v₁, v₂, v₃} taking the place of the familiar x, y and z directions.

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