1.1.1     Given a linear system, specify the coefficient matrix and the augmented matrix of the system.
             (1.1 Exercises: 1-4)

1.1.2     Solve linear systems.  (Exercises 1.1:  11-14)

1.1.3     Given a linear system, apply elementary row operations to the point where you can determine
             whether or not the system is consistent, and if it is consistent determine whether or not the
             solution is unique.  (1.1 Exercises: 15, 16)

1.1.4     Explain when a matrix is in (upper) triangular form.

1.2.1     Define what it means to say a matrix is in reduced (row) echelon form.

1.2.2     Show how to reduce a matrix to reduced (row) echelon form.  (1.2 Exercises: 3, 4)

1.2.3     Identify the pivot positions and columns of a matrix. (1.2 Exercises: 3, 4)

1.2.4     Apply the row reduction algorithm to the augmented matrix of a linear system to
             Produce the unique reduced echelon form of that matrix.  (1.2 Exercise 5)

1.2.5     Given the reduced echelon form for the augmented matrix of a linear system, Determine whether
             or not the system is consistent, and if it is consistent specify either the unique solution or the
             general form for all solutions.

1.2.6     State the Existence and Uniqueness Theorem for linear systems.

1.3.1     Define addition of column vectors and multiplication of a column vector by a scalar.

1.3.2     Provide a geometric interpretation of (column) vector addition and scalar multiplication.

1.3.3     State the algebraic properties of vector addition and scalar multiplication in Rⁿ.

1.3.4     State the definition of linear combination.

1.3.5     Given two non-zero vectors in R³ and a third vector in R³, determine whether or not the third
             vector can be written as a linear combination of the two given vectors.  If the third vector is a
             linear combination of the given two vectors, write out that linear combination.  (1.3 Exercises: 11, 12)

1.3.6     Given sketches two non-zero vectors in R² and a third vector in R², determine whether or not the
             third vector can be written as a linear combination of the two given vectors.  If the third vector is a
             linear combination of the given two vectors, sketch the parallelogram whose diagonal is the third
             vector.  (1.3 Exercises: 7, 8)

1.3.7     Given vectors v₁, v₂,…, v_p in Rⁿ, define the subset of Rⁿ spanned (or generated) by v₁, v₂, …,v_p.

1.3.8     Given vectors v₁, v₂,…, v_p in R³, and another vector b in R³, determine whether or not b is in
             Span{ v₁, v₂,…, v_p}.  (1.3 Exercises: 13, 14, 17, 25)

1.4.1     Define the product of an m x n matrix A and a column vector x in Rⁿ.

1.4.2     Given an m x n matrix A and a vector b, find Ab.  (1.4 Exercises:  1-4)

1.4.3     Given a linear combination of  vectors, write the linear combination as matrix times a vector.
             (1.4 Exercises: 7, 8)

1.4.5     Given a matrix equation, write the equation as a vector equation.  (1.4 Exercises: 5-6)

1.4.6    Solve matrix equations Ax = b.  (1.4 Exercises:  11-13)

1.4.6     Given a matrix A with m rows for a specified m, determine whether or not the columns of A span R^m.
             (1.4 Exercises:  18, 19, 37)

1.4.7     Given a set of vectors {v₁, v₂,…, v_p } in R^m, determine whether or not  Span{ v₁, v₂,…, v_p} = R^m.
             (Exercises 1.4:  21, 22)

1.5.1     Given a homogeneous equation Ax = 0.  Solve the system.  If the system has infinitely many
             solutions, specify the general form of the solutions in parametric vector form.  (Exercises 1.5:  7-12)

1.5.2     Given a consistent linear system with infinitely many solutions, specify the general form of the
             solutions in parametric vector form.  (1.5 Exercises:  15, 16)

1.7.1     Define what it means for an indexed set of vectors { v₁, v₂,…, v_p} in Rⁿ to be linearly
             independent.

1.7.2     Determine whether or not a particular indexed set of vectors {v₁, v₂,…, v_p } is linearly
             independent.  (Exercises 1.7: 1-4, 15-20)

1.7.3     Determine whether or not the columns of a particular matrix A are linearly independent.
             (1.7 Exercises: 5-8)

1.7.4     Given a lineraly dependent set of vectors {v₁, v₂,…, v_p }, find one linear dependence relation
             among v₁, v₂,…, v_p.

1.8.1     Define:  transformation (or function or mapping) T from Rⁿ to R^m.

1.8.2     Given T from Rⁿ to R^m, define the domain, codomain, and range of  T.

1.8.3     Define: linear transformation

1.8.4     Given a rule for T from Rⁿ to R^m, determine whether or not T is a linear transformation.
             (1.8 Exercises: 29, 30, 32, 33, 35, 36)

1.8.5     Given T from Rⁿ to R^m, defined by T(x) = Ax where A is a specified m x n matrix, and  specified
             vectors u, b, and c, (a) Find T(u), (b) Find an x in R^m such that T(x) = b, (c) Determine whether
             or not c is in the range of T.  (1.8 Exercises: 1-6,  9-12)

1.9.1     Given a linear transformation T from Rⁿ to R^m, find the standard matrix for the linear
             transformation.  (1.9 Exercises:  1-10, 17-20)

1.9.2     Define what it means to say T from Rⁿ to R^m is onto R^m.

1.9.3     Define what it means to say T from Rⁿ to R^m is 1-1.

1.9.4     Given T from Rⁿ to R^m, determine whether or not T  is (a) onto R^m, (b) 1-1.  (1.9 Exercises: 25-28)

1.9.5     Demonstrate your knowledge of the terms, concepts, and properties of  Chapter 1 by responding
             to true-false items.  Justify your “false” responses.  (1.1 Exercises: 23, 24; 1.2 Exercises: 21, 22;
             1.3 Exercises: 23, 24; 1.4 Exercises: 23, 24; 1.5 Exercises: 23, 24; 1.7 Exercises: 21, 22, 33-38;
             1.8 Exercises: 21-22; 1.9 Exercises: 23, 24)

2.1.1    Calculate sums, scalar products, and matrix products of matrices.  (2.1 Exercises:  1-11)

2.1.2    Find the transpose of a given matrix, and demonstrate knowledge of properties of the
            transpose operation.

2.1.3    Demonstrate knowledge of the properties of matrix operations.  (2.1 Exercises: 15, 16)

2.2.1    Given an n x n matrix, determine whether or not the matrix is nonsingular, and if it is find its
            inverse.  (Exercises 2.2:  1-4, 29-33)

2.2.2    Demonstrate the ability to make judgements about the truth of generalizations about singular
            and nonsingular matrices.  (Exercises 2.2:  9-10)

2.3.1    Demonstrate the ability to apply The Invertable Matrix Theorem (p. 129).
            (Exercises 2.3: 1-8, 11-32, 35)

2.3.2    Relate the concepts of "matrix inverse" and the "inverse of a linear transformation."
            (Exercises 2.3: 33, 34)

2.8.1    Define:  subspace.

2.8.2    Given a subset S of Rⁿ , for some n, determine whether or not S is a subspace of Rⁿ and
            justify your answer.  (2.8 Exercises:  1-4)

2.8.3    Define:  column space of a matrix A

2.8.4    Define:  null space of a matrix A

2.8.5    Define:  basis for a subspace S of Rⁿ

2.8.6    Given a subspace, determine whether or not a specified vector is in the subspace.
            (2.8 Exercises:  5-10)

2.8.7    Specify the standard basis for Rⁿ for a particular value of n.

2.8.8    Given a subspace S of Rⁿ for some particular value of n, find a basis for S.
            (2.8 Exercises:  15-20, 23-26)

2.8.9   Respond to true-false items related to the concepts of subspace, column space, basis.
            (2.8 Exercises: 21, 22)

2.9.1    Given a basis B for a subspace S and a vector v in S, find both the coordinates of
            v relative to the basis B and the coordinate vector of v relative to B.
            (2.9 Exercises:  1-7 odd)

2.9.2    Define:  dimension of a subspace

2.9.3    Given a subspace, determine its dimension.

2.9.4    Define:  rank of a matrix A

2.9.5    Given a matrix, determine its rank.

2.9.6    State:  The Rank Theorem

2.9.7    State:  The Basis Theorem

2.9.8    Demonstrate knowledge of the concepts and properties of Chapter 2.
            (2.9 Exercises:  17-24;  Chapter 2 Supplementary Exercises:  1)

4.1.1    Define:  vector space

4.1.2    Define:  subspace

4.1.3    Define: Span{v₁, v₂, ..., v_p}

4.1.4    Given a set of objects upon which is defined addition and multiplication by real numbers,
            determine whether or not the set is a vector space.  (4.1 Exercises: 1, 2, 3)

4.1.5    Given a vector space V and a subset S of V.  Determine whether or not S is a subspace of V.
            (4.1 Exercises:  4, 5-18)

4.2.1    Define:  null space of an m x n matrix A.

4.2.2    Define:  column space of an m x n matrix A.

4.2.3    Define:  kernel of a linear transformation

4.2.4    Given a matrix A, find an explicit description of Nul A.  (4.2 Exercises:  3-6)

4.2.5    Given a set W, determine whether or not the set is a vector space.  (4.2 Exercises:  7-14)

4.2.6    Prove:  The range of any linear transformation from a vector space V into a vector
            space W is a subspace of W.

4.2.7    Given a linear transformation T from a vector space V into a vector space W,
            find Ker T.

4.3.1    Define:  basis of a subspace of a vector space

4.3.2    Given a matrix A, specify bases for Nul A and Col A.  (4.3 Exercises: 13, 14)

4.3.3    Given a set of vectors that span a subspace, find a basis for that subspace.
            (4.3 Exercises: 15-18)

4.4.1    Define:  coordinates of a vector x relative to the basis B

4.4.2    Prove:  If B = {v₁,..., v_n} is a basis for a vector space V, then the coordinate mapping
             x to [x]_B  is a 1-1 linear transformation from V onto Rⁿ.

4.4.3    Given a basis B and the coordinate vector [x]_B, find the coordinates of x relative to the
            standard basis.  (4.4 Exercises: 1-4)

4.4.4    Given a vector x and a basis B, find [x]_B.  (4.4 Exercises: 5-8, 13)

4.4.5    Given a basis B, find the change-of-coordinates matrix from B to the standard basis.
            (4.4 Exercises: 9, 10)

4.5.1    Define:  dimension of a vector space

4.5.2    State:  The Basis Theorem

4.5.3    Given a description of a subspace, find a basis for the subspace and specify the
            subspace's dimension.  (4.5 Exercises:  1-18)

4.6.1    Define:  the rank of a matrix A

4.6.2    State:  The Rank Theorem

4.6.3    Given a matrix A, specify rank A, and dim Nul A, and then find bases for Col A,
            Row A, and Nul A.  (4.6 Exercises: 1-4)

4.7.1    Given bases B and C for a vector space V, find the change-of-coordinates matrix
            from B to C and the change-of-coordinates matrix from C to B.
            (4.7 Exercises:  1, 2, 5, 7-10)

4.7.2    Given bases B and C for a vector space V and [x]_B for some x in V, find [x]_C.
            (4.7 Exercises: 1, 2, 5, 6)

4.*.1    Demonstrate knowledge and understanding of the terms, concepts and theory of Chapter 4
            in responding to TRUE-FALSE items and justifying all "FALSE" responses.
            (4.1 Exercises:  23, 24; 4.2 Exercises: 25, 26; 4.3 Exercises: 21, 22; 4.4 Exercises: 15, 16;
            4.5 Exercises: 19, 20, 29, 30; 4.6 Exercises: 17, 18)

5.1.1    Define:  eigenvector, eigenvalue

5.1.2    Given a matrix A and an eigenvalue lambda, find a corresponding eigenvector.
            (5.1 Exercises: 7, 8)

5.1.3    Given a matrix A and an eigenvalue lambda, explicitly describe the corresponding eigenspace
            and find a basis for that eigenspace.  (5.1 Exercises:  9-16)

5.1.3    Given a 2 x 2 or 3 x 3 matrix, find its eignevalues.  (5.1 Exercises: 17, 18)

5.2.1    Given a 2 x 2 or 3 x 3 matrix specify the characteristic polynomial and characteristic equation
            for the matrix and calculate the eigenvalues of the matrix.  (5.2 Exercises:  1-8)

5.3.1    Given a square 2 x 2 or 3 x 3 matrix A and a matrix P such that A = PAP^-1, find a formula

            for A^k.

5.3.2     Given a 2 x 2 or 3 x 3 matrix A, if possible diagonalize A.

5.4.1    Given a linear transformation T from a vector space V of dimension 2 or 3 into a vector space

            W of dimension 2 or 3 and a basis B for W, find the matrix for T relative to the standard basis

            for V and the basis B for W.    

5.4.2    Given a linear transformation T on R² or R³ and a basis B, find the matrix representation for T relative to B. 

6.1.1    Given vectors x and y, calculate each of the following:  x · y, ||x||, ||x - y||, and ||(1/||x||)x||.

6.1.2    Determine whether or not a pair of vectors are orthogonal.

6.1.3    Determine whether or not each of the vectors in one subspace are orthogonal to each of the vectors in another subspace.

6.2.1    Define:  orthogonal set

6.2.2    Define:  orthogonal basis

6.2.3    Define:  orthonormal set

6.2.4    Define:  orthonornal basis

6.2.5    Given a basis for a vector space determine whether or not it is an orthogonal basis.

6.2.6    Given a basis for a vector space determine whether or not it is an orthonormal basis.

7.1.1    Orthogonally diagonalize a 2 x 2 or 3 x 3 symmetric matrix given its eigenvalues. 

            Provide an orthogonal matrix P  and a diagonal matrix D.

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