MAT 335 Review Guide for Test 1 Fall 2013The test will cover material discussed in class and studied in exercises, from Section 1.1 toSection 2.3. This guide provides some suggestions and notes some of the most importantdefinitions, theorems, and skills from these sections.Suggestions• Read the text book and review the class notes.• Review/redo the homework. Please pay special attention to the ungraded problemsin the homework.• Make your own summary.• Study and review in a group.Review SessionTime: Wednesday 10/02, 7:00 PM,Place: BR 101DefinitionsThe definition of A~x in both words and symbols.Row Reduction Echelon Forms.Pivot positions and pivot columns.Span{~v}, Span{~u, ~v} and geometric interpretation in R2or R3.Span{~v1, . . . , ~vp}.Linearly independent and linearly dependent.Linear transformation.One-to-one and onto (for linear transformations).The definition of a matrix product AB.The definition of a transpose of a matrix A.The definition of “invertible matrix”.SkillsMAT 335 Review Guide for Test 1 Fall 2013• Perform elementary row operations.• Find the echelon/reduced echelon form of a matrix.• Determine when a system is consistent. Write the general solution in parametricvector form.• Determine values of parameters that make a system consistent or make the solutionunique.• Describe existence or uniqueness of solutions in terms of pivot positions.• Determine when a homogeneous system has a nontrivial solution.• Determine when a vector is in a subset spanned by specified vectors.• Exhibit a vector as a linear combination of specified vectors.• Determine whether the columns of an m × n matrix span Rm.• Determine whether the columns are linearly independent.• Determine whether a set of vectors is linearly independent. Know several methodsthat can sometimes produce an answer without much calculation.• Determine if a transformation is linear.• Find the standard matrix of a linear transformation.• Determine whether a specified vector is in the range of a linear transformation.• Determine whether a linear transformation T : Rn→ Rmis one-to-one or maps Rnonto Rm.• Perform standard matrix operations.• Find the inverse of a matrix using by row reducing [A I] to [I A−1].• Use an inverse matrix to solve a system of linear equations.• Use matrix algebra to solve equations involving matrices.Theorems in Chapter 1Theorem 2 (Existence and Uniqueness Theorem)A linear system is consistent if and only if the rightmost column of the augmented matrixMAT 335 Review Guide for Test 1 Fall 2013is not a pivot column – that is, if and only if an echelon form of the augmented matrix hasno row of the form[0 · · · 0 b] with b nonzero .If a linear system is consistent, then the solution set contains either (i) a unique solution,when there are no free variables, or (ii) infinitely many solutions, when there is at leastone free variable.Theorem 3 (Matrix equation, vector equation, system of linear equations)If A is an m × n matrix, with columns ~a1, . . . ,~an, and if~b is in Rm, the matrix equationA~x =~bhas the same solution set as the vector equationx1~a1+ x2~a2+ · · · + xn~an=~bwhich, in turn, has the same solution set as the system of linear equations whose augmentedmatrix is[~a1~a2· · · ~an~b].Theorem 4(Conditions for columns of A to span Rm)Let A be an m × n matrix. Then the following statements are logically equivalent. Thatis, for a particular A, either they are all true statements or they are all false.(a) For each~b in Rm, the equation A~x =~b has a solution.(b) Each~b in Rmis a linear combination of the columns of A.(c) The columns of A span Rm.(d) A has a pivot position in every row.Theorem 5(Properties of the Matrix-Vector Product A~x)If A is an m × n matrix, ~u and ~v are vectors in Rn, and c is a scalar, then:(a) A(~u + ~v) = A~u + A~v;(b) A(c~u) = c(A~u).Theorem 7 (Characterization of Linearly Dependent Sets)An indexed set S = {~v1, . . . , ~vp} of two or more vectors is linearly dependent if and only ifat least one of the vectors in S is a linear combination of the others. In fact, if S is linearlyMAT 335 Review Guide for Test 1 Fall 2013dependent and ~v16=~0, then some ~vj(with j > 1) is a linear combination of the precedingvectors, ~v1, . . . , ~vj−1.Theorem 8 (Properties of linearly dependent sets)If a set contains more vectors than there are entries in each vector, then the set is linearlydependent. That is, any set {~v1, . . . , ~vp} in Rnis linearly dependent if p > n.Theorem 9 (Properties of linearly dependent sets)If a set S = {~v1, . . . , ~vp} in Rncontains the zero vector, then the set is linearly dependent.Theorem 11 (one-to-one linear transformations)Let T : Rn→ Rmbe a linear transformation. Then T is one-to-one if and only if theequation T (~x) =~0 has only the trivial solution.Theorem 12 (onto linear transformations)Let T : Rn→ Rmbe a linear transformation and let A be the standard matrix for T. Then:(a) T maps Rnonto Rmif and only if the columns of A span Rm;(b) T is one-to-one if and only if the columns of A are linearly independent.Theorems in Chapter 2Theorem 4Let A =a bc d. If ad − bc 6= 0, then A is invertible andA−1=1ad − bcd −b−c a.If ad − bc = 0, then A is not invertible.Theorem 5If A is an invertible n × n matrix, then for each~b in Rn, the equation A~x =~b has theunique solution ~x = A−1~b.Theorem 6(a) If A is an invertible matrix, then A−1is invertible and (A−1)−1= A.(b) If A and B are n × n invertible matrices, then so is AB, and the inverse of AB is theproduct of the inverses of A and B in the reverse order. That is, (AB)−1= B−1A−1.MAT 335 Review Guide for Test 1 Fall 2013(c) If A is an invertible matrix, then so is AT, and the inverse of ATis the transpose ofA−1. That is, (AT)−1= (A−1)T.Theorem 7An n × n matrix A is invertible if and only if A is row equivalent to In, and in this case,any sequence of elementary row operations that reduces A to Inalso transforms InintoA−1.Theorem 8 (The Invertible Matrix Theorem)Let A be a square n × n matrix. Then the following statements are equivalent. That is,for a given A, the statements are either all true or all false.(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.(d) The equation A~x =~0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation ~x 7→ A~x is one-to-one.(g)