Due Wednesday 11/7 at 8am

Section 5.2: The Characteristic Equation

Reading questions:

1. Let A=. Find the characteristic equation of A.

   I say:
   

   The characteristic equation is det(A-(lambda)I)=0. A-(lambda)I has 7-lambda and 1-lambda on the diagonal, and is otherwise unchanged.

   Thus det(A-(lambda)I)=(7-lambda)(1-lambda)+8, and therefore, the characteristic equation is
   (7-lambda)(1-lambda)+8=0.

2. How is the characteristic equation of a matrix related to the eigenvalues of t he matrix?

   JL, RZ, and JA combine to say:
   

   They are related because lambda is an eigenvalue of an n X n matrix A if and only if lambda satisfies the characteristic equation det(A- lambda *I) =0.

   Characteristic equations make it so that there is only one variable in the (A-kI)x=0 equation.

   The multiplicity of an eigenvalue lambda is its multiplicity as a root of the characteristic equation.

Due Monday 11/5 at 8am

Section 5.1: Eigenvectors and Eigenvalues

Reading questions:

Suppose A is a 3 x 3 matrix with eigenvalues 1, 2, and 5.
1. What is the dimension of Nul(A)?

   RZ says:
   

   The dimension of Nul(A) is 0 becasue of the Invertible Matrix Theorem.

   I add:
   

   A is a square matrix and 0 is not an eigenvalue. That means that A is invertible by the final statement of the invertible matrix theorem. Hence A has a pivot in every column. Since dim(Nul(A))=the number of free variables, dim(Nul(A))=0.

2. What is the dimension of the eigenspace of A?

   MB says:
   

   The dimension of the eigenspace of A is 3.

   I add:
   

   For each eigenvalue, there exists an eigenvector. Since the eigenvalues are distinct, Theorem 2 tells us that these three eigenvectors are linearly independent. That meanst that dim(eigenspace(A)) is at least 3. But since the eigenvectors are in R^3, the dimension of the eigenspace must be no more than 3. Hence the dimension of the eigenspace is exactly 3. -->

Due Friday 11/2 at 8am

Section 4.6: Rank
Introduction to Chapter 5
Section 5.1: Eigenvectors and Eigenvalues

Reading Questions:

1. If A is a 4 x 7 matrix with three pivots, what is the dimension of Nul A? Why?

   RZ says:
   

   The dimension of NulA when A is a 4x7 matrix with 3 pivots is 4 because the dimension of NulA is the number of free variables in a matrix. So this means if A has 3 pivots then there must be 4 columns without pivots making them free variables.

   GC says:
   

   The dimension of Nul A is four. By Theorem 13, the number of columns in a matrix equals the number of pivots plus the dimension of Nul A. 3+4=7

2. Let A= . Verify that (1, -2) is an eigenvector of A with corresponding eigenvalue 3.

   MT says:
   

   A= [(7, 2)], [(-4,1)] and u= (1, -2).

   We multiply Au and get the vector (3, -6), which can be reduced to 3(1, -2).

   (Therefore, it is true that Au=3u,), and so (1, -2) is an eigenvector and 3 is an eigenvalue.

Due Wednesday 10/31 at 8am

Section 4.5

Reading questions:

1. What is the dimension of R³? Why? Does this make sense geometrically?

   MT says:
   

   Seeing as how the basis for Rⁿ has n vectors, the dimension of Rⁿ must be n, and therefore the dimension of R³ is 3. This makes sense geometrically because R³ is the 3rd dimension, go figure!

   JL adds:

   Of course this makes geometric sense because there are three different direction sets.

2. Can there be a set of linearly independent vectors {v₁, v₂, ..., v₁₂} that does not span R¹²? Explain.

   RZ says:
   

   This question does not make sence to me but I might have missed something in the reading.

   If you meant to have the set {v1, v2, ..., v12} then it must span R¹², because according to Theorem 12 (The Basis Theorem), the set is automatically a basis for R¹², which means that the set spans R¹².

Due Friday 10/26 at 8am

Section 4.3: Linearly Independent Sets; Bases

Reading questions:

1. Let v₁=(1,2), v₂=(3,4), and v₃=(4,6). Give a basis for H=Span{v₁, v₂, v₃}.

   I say:
   

   Since v₃=v₁+v₂, v₃ is a linear combination of v₁ and v₂, and so is redundant.

   Thus we can span the same space with just v₁ and v₂. Now the question is, can we eliminate another one?

   When you're only dealing with two vectors, the only way they can be linearly dependent is if one is a multiple of the other. Since that is not the case here, a basis for H is {v₁, v₂}.

2. If A is a 4 x 5 matrix with three pivot positions, how many vectors does a basis for Col A contain?

   MT says:
   

   According to theorem 6, the pivot columns of a matrix A form a basis for Col A. A is a 4x5 matrix with 3 pivot positions so three vectors form a basis for ColA.

Due Wednesday 10/24 at 8am

Section 4.2: Null Spaces, Column Spaces, and Linear Transformations

Reading questions:

1. True or False: If A is a 3 x 5 matrix, then Nul A and Col A are subspaces of R³
   GC says:
   

   False. In this case, Nul A is a subspace of R5, and Col A is a subspace of R3.

2. Let A=. Find Nul A.
   Combining the answers of JM and RZ:

   2x₁=0, 4x₂=0
   => Nul A is {(0,0)}

Due Monday 10/22 at 8am

Introduction to Chapter 4
Section 4.1: Vector Spaces and Subspaces

Reading questions:

1. Is the subset of R³ consisting of all scalar multiples of the vector (5, 6, -3) a subspace of R³? Why or why not?

   SB says:
   

   The subset of R³ that consists of the all the scalar multiples of the vector (5, 6, -3) is a subspace of R³ because it passes through the zero vector and it is closed under both vector multiplication and addition.

2. Give an example of a subset of R² that is not a subspace of R²?

   RZ says:
   

   An example of a subset of R2 that is not a a subspace of R2 is a line that does not go through (0,0).

Due Wednesday 10/17 at 8am

Introduction to Chapter 3
Section 3.1: Introduction to Determinants
Section 3.2: Properties of Determinants

Reading questions:

1. Why do we care about finding det(A)?

   MT says:
   

   A determinant can give us formulas for the volume of different polyhedra. They also give us information about matrices, information that is very useful in linear algebra.

   MB adds:
   

   If det(A) does not equal zero and is a square matrix, then it is invertible.

2. If A=, what is det(A)?

   SB says:
   

   A cofactor expansion down the first column will make
   detA=2*matrix( [ [ 4, -4], [0, 1] ]) = [=2*(4*1-(-4)*0) ] = 2*4=8

Due Monday 10/15 at 8am

Section 2.8: Applications to Computer Graphics

Reading questions:

1. What is the advantage of using homogeneous coordinates in computer graphics?

   GC says
   

   Using homogeneous coordinates is advantageous because it allows the mathematician to use a matrix to represent a translation.

   SB adds:
   

   This is especially useful in biology where biologists will use computer graphics to similate a protein. The visualizing of potential chemical reactions and changes in view can be done by using homogeneous coordinates.

Due Friday 10/12 at 8am

Section 1.9: Linear Models in Business, Science, and Engineering
Section 4.9: Applications to Markov Chains

Reading questions:

1. What is a steady state vector for a stochastic matrix P?

   JM says
   

   A steady state vector for a stochastic matrix P is a probability vector such that Pq=q.

2. What is special about regular stochastic matrices?

   SB says:
   

   Each regular Stochastic Matrix has a steady state vector.

Due Wednesday 10/10 at 8am

Section 1.9: Linear Models in Business, Science, and Engineering
Section 4.9: Applications to Markov Chains

Reading questions:

1. What is the point of studying Markov chains?

   MB says:
   

   The point of studying Markov chains is to take current trends of a specific situation and use this to predict what will happen in the future.

   MT says:
   

   Markov chains are a way for the Linear Algebra we have been doing to be applied to everyday situations. The way that they are set up enables us to predict certain outcomes with specific results. If we know what x0 is, then the nature of the chain allows us to find xk.

   JL says:
   

   The point in studying Markov chains is that they are used in many fields. The examples from the book are biology, business, chemistry, engineering, and physics. They are good at representing the data of an experiment or measurement.

Due Wednesday 10/3 at 8am

Section 2.2 : The Inverse of a Matrix
Section 2.3: Characterization of Invertible Matrices

Reading Questions:

1. Find the inverse of the matrix A defined as . Use Maple notation to write the result.

   SB says:
   

   The inverse of matrix A is
   Matrix([[1/6, -2, -9], [0, 1/2, 2], [0, 0, 1]]); or

   æææThis was attained by (augmenting) matrix A with the Identity matrix as shown in example 7 of Section 2.2, and row reducing the far left side, until the identity matrix could be seen in the place of where matrix A was. æThat is, reducing it to reduced echelon form on the far left.

2. Suppose A=[u, v, w] is invertible. What is span{u, v, w}?

   
   GC says:
   

   The span of {u, v, w} is R3, according to Theorem 8.

   I add:
   

   This question is so simply asked, and yet it is surprisingly layered. Since A is invertible, we know it is square. Since there are 3 columns, there must be 3 rows, and hence u, v, and w are all in R³. Also because A is invertible,we know that there's a pivot in every row. Thus we know that Span(u,v,w)=R³.

3. In Example 2 in Section 2.3, how do we know that T maps Rⁿ onto Rⁿ?

   MT says:
   

   The Invertible Matrix Theorem points out that either all of its statements are true or all of it's statements are false. So if A is invertible, then by the IMT, the linear transformation T maps Rn onto Rn

   I add:
   

   To understand why this is true, we can go through some of the logic: T is one-to-one, so the columns of the associated matrix A are linearly independent, and so A has a pivot in every column. Because T goes from R^n, A has n columns and so A has n pivots. But since T goes to R^n, A also has n rows, and so there's a pivot in every row and thus T is onto.

Due Monday 10/1 at 8am

Introduction to Chapter 2
Section 2.1: Matrix Operations
Section 2.2 : The Inverse of a Matrix

Reading questions:

1. If A is a 10 x 7 matrix and B is a 2 x 10 matrix, does AB exist, and if so, how many columns does it have? How about BA?

   GC says:
   

   AB does not exist, because the 7 columns of A do not match the 2 rows of B, and it is impossible to multiply the two matrices together.

   However, BA does exist, and by definition, it has 7 columns.

2. Give one way in which matrix multiplication differs from multiplication of real numbers.

   JM says:
   

   Matrix multiplication differs from multiplication of real numbers in that the cancellation laws do not hold true, for example, if AB=AC, B=C may not be true.

   RZ adds:
   

   (Another) way that matrix multipllication differs from multiplication of real numbers is that it is not commutative: AB does not always equal BA.

   JL finishes with:
   

   (Finally), multiplying matrices is different then multiplying regular numbers because with matrices, it may be possible to multiply x by y but not y by x, where as with real numbers, it doesn't matter which order you put them in.

3. Suppose A is an invertible matrix. Can Ax=b have infinitely many solutions?

   BH says:
   

   No, it does not have an infinite number of solutions. It has the unique solution x=A^-1 b, by theorem 5.

Due Friday 9/28 at 8am

Section 1.8: The Matrix of a Linear Transformation

Reading questions:

Let T:R⁵ --> R³ be a linear transformation with standard matrix A, where A has three pivots.
1. Is T one-to-one?

   
   

   T is not one-to-one, because of the existence of free variables. By definition, T is one to one only if each b in Rm has either a unique solution (to T(x)=b) or none at all.

2. Is T onto?

   BH says:
   

   Since there is a pivot in evey row, A spans R^m which makes T onto, because of theorem 12.

Due Wednesday 9/26 at 8am

Section 1.8: The Matrix of a Linear Transformation

Reading questions:

1. Give the matrix A for the linear transformation T: R² --> R² that rotates the plane by Pi/4 degrees counter-clockwise.

   SB says:
   
   

   matrix( [ [1], [ 0] ]) rotates onto matrix([ [cos(pi/4)] , [sin(pi/4) ] ]).
   matrix( [ [0], [ 1] ]) rotates onto matrix([ [-sin(pi/4)], [cos(pi/4) ] ]).

   Therefore A = matrix( [ [cos(pi/4), -sin(pi/4)], [ sin(pi/4) , cos(pi/4)] ¾

2. Give the matrix A for the linear transformation T:R² --> R² that expands horizontally by a factor of 2.

   RZ says:
   

   Matrix([[2, 0], [0, 1]]) is the matrix A for the linear transformation T: R2--> R2 that expands horizontally by a factor of 2.

Due Monday 9/24 at 8am

Section 1.7: Introduction to Linear Transformations

Reading Questions:
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