1.1

1. A 5X6 matrix has six rows. False. 5 Rows.2. The solution set of a linear system involving variables x1,...,xn is a list of

numbers (s1,...,sn) that makes each equation in the system a true statement when the values s1,...,sn are substituted for x1,...,xn, respectively. False. Description applied only works for one given solution, not necessarily all given solutions.

3. Two matrices are row equivalent if they have the same number of rows. False. Sequence of row operations that transform one matrice into another.

4. An inconsistent system has more than one solution. False. Has 0 solutions.

1.2

1. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. False. Each matrix is row equivalent to only one reduced echelon matrix.

2. If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent. False. The row shown only says that 5x4 = 0. Does not reveal if it is consistent or inconsistent.

3. The row reduction algorithm applies only to augmented matrices for a linear system. False. Applies to any matrix.

4. The echelon form of a matrix is unique. False. Only the reduced echelon form is unique.

5. The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. False. Pivot positions in a matrix are solely determined by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix.

6. Whenever a system has free variables, the solution set contains many solutions. False. The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, solution set is empty.

1.3

1. Another notation for the vector [-4/3] is [-4 3]. False. (-4, 3)2. The points in the plane corresponding to [-2/5] and [-5/2] lie on a line

through the origin. They have to be multiples to go through the origin.3. The set Span {u,v} is always visualized as a plane through the origin. False.

Span {u,v} is not a plane when v is a multiple of u or when u is the 0 vector.4. The weights c1,...,cp in a linear combination c1v1+...+cpvp cannot all be

zero. False. Can be any linear combination including zeros.

1.4

1. The equation Ax = b is referred to as a vector equation. False. Referred to as matrix equation.

2. The equation Ax = b is consistent if the augmented matrix [A b]has a pivot position in every row. False. It is about a coefficient matrix, not an augmented matrix.

3. If the augmented matrix [A b]has a pivot position in every row, then the equation Ax = b is inconsistent. False. It is about a coefficient matrix, not an augmented matrix.

1.5

1. The solution Ax = 0 gives an explicit description of its solution set. False. Gives an implicit, not explicit description.

2. The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. False. Equation Ax = 0 always has the trivial equation.

3. The equation x = p + tv describes a line through v parallel to p. False. Describes line p parallel to v.

4. The solution set of Ax = b is the set of all vectors of the form w = p + vh where vh is any solution of the equation Ax = 0. False. Solution set could be empty, statement only true when there exists vector p so that ap = b.

5. If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero. False. Nontrivial solution is any nonzero x that satisfies the equation.

6. The solution set of Ax = b is obtained by translating the solution sex of Ax = 0. False. Statement only true when Ax = 0 is nonempty.

1.7

1. The columns of a matrix A are linearly independent if the equation Ax = 0 has the trivial solution. False. Homogenous system always has trivial solution.

2. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in s. False. Does not that say that every vector is necessarily in a linearly dependent set is a linear combination of the preceding vectors.

3. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. False. Nothing is said in regards to when number of vectors in the set does not exceed number of entries in each vector.

4. If a set in R^N is linearly dependent, then the set contains more vectors than there are entries in each vector. False. There can be equal vectors and entries in each vector.

1.8

1. If A is a 3X5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R^3. False. The domain is r^5.

2. If A is an mxn matrix, then the rage of the transformation x-->Ax is R^m. False. The range is the set of all linear combinations of the columns of A.

3. Every linear transformation is a matrix transformation. False. Every matrix transformation is a linear transformation.

4. The codomain of the transformation x--> Ax is the set of all linear combinations of the columns of A. False. If A is in mxn matrix, codomain is r^m.

5. If T: R^n --> r^m is a linear transformation and if c is in R^m, then a uniqueness question is "Is c in the range of T?" False. The question is an existence question, not uniqueness.

1.9

1.2. When two linear transformations are performed one after another, the

combined effect may not always be a linear transformation. False. Will be a linear transformation, but turns into a question of existence and uniqueness.

3. A mapping T: R^n-->R^m is onto R^m if every vector x in r^n maps onto some vector in r^m. False. Any vector mapped from r^n to r^m will map one vector to another vector.

4. If A is a 3x2 matrix, then the transformation x--> Ax cannot be one-to-one. False. For instance, A can be the combination of the 3x2 matrix, and followed by [x1/x2], showing linear transformation.

5. Not every linear transformation from r^n to r^m is a matrix transformation. False. Every linear transformation from r^n to r^m is a matrix transformation.

6. The standard matrix of a linear transformation from r^2 to r^2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form [a/0 0/d], where a and d are +- 1. False. Any function from r^n to r^m maps a vector onto a single unique solution.

2.1

1. If A and B are 2x2 with columns a1, a2, and b1, b2, respectively, then AB = [a1b1 a2b2]. False. AB = ab1 ab2 etc...

2. Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A. False. The B and A in second part of statement should be swapped.

3. The transpose of a product of matrices equals the product of their transposes in the same order. False. Not in the same order, but should be worded to in the reverse order.

4. If A and B are 3x3 and B = [b1 b2 b3], then AB = [ab1 + ab2 + ab3]. False. Plus signs should be just spaces if you want to make the statement true.

5. (AB)C = (AC)B. False. The order cannot be changed. Should be A(BC) = (AB)C.

6. (AB)^T = A^TB^T. False. Should be B^TA^T.