Solutions to Homework 2 - Math 3410 1. (Page 156: # 4.72) Let V be
... Show that V satisfies all the axioms of a vector space except [M4 ], that is, except 1~u = ~u. Hence [M4 ] is not a consequence of the other axioms. Solution The proofs of [A1 ], [A2 ], [A3 ], and [A4 ] are straightforward. Note that ~0 = (0, 0) and −(a, b) = (−a, −b). We have k((a1 , b1 ) + (a2 , b ...
... Show that V satisfies all the axioms of a vector space except [M4 ], that is, except 1~u = ~u. Hence [M4 ] is not a consequence of the other axioms. Solution The proofs of [A1 ], [A2 ], [A3 ], and [A4 ] are straightforward. Note that ~0 = (0, 0) and −(a, b) = (−a, −b). We have k((a1 , b1 ) + (a2 , b ...
Math:HS Number and Quantity
... a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy). b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when ...
... a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy). b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when ...
Revisions in Linear Algebra
... A matrix is a rectangular table of (real or complex) numbers. The order of a matrix gives the number of rows and columns it has. A matrix with m rows and n columns has order m × n. Write down the order of the following matrices: ...
... A matrix is a rectangular table of (real or complex) numbers. The order of a matrix gives the number of rows and columns it has. A matrix with m rows and n columns has order m × n. Write down the order of the following matrices: ...
p:texsimax -1û63û63 - Cornell Computer Science
... This is the Sherman–Morrison formula and it can be used to compute (A + uv T )−1 from A−1 in O(n2 ) flops. See [2, p. 50]. In a linear equation setting, the Sherman– Morrison result, together with a QR factorization of A, makes it possible to solve (A + uv T )x = b in O(n2 ) flops. Interest in a “fast ...
... This is the Sherman–Morrison formula and it can be used to compute (A + uv T )−1 from A−1 in O(n2 ) flops. See [2, p. 50]. In a linear equation setting, the Sherman– Morrison result, together with a QR factorization of A, makes it possible to solve (A + uv T )x = b in O(n2 ) flops. Interest in a “fast ...
Exam 1 Solutions
... Directions: Do ALL of your work on these sheets of paper. Show all your steps. No calculators are allowed. Remember, the honor code is to be observed on this exam. You are allowed one study sheet on this exam. Suggestion: some of the problems say: show something. Be guided in your answer by how much ...
... Directions: Do ALL of your work on these sheets of paper. Show all your steps. No calculators are allowed. Remember, the honor code is to be observed on this exam. You are allowed one study sheet on this exam. Suggestion: some of the problems say: show something. Be guided in your answer by how much ...
LINEAR DEPENDENCE AND RANK
... Another theorem of linear algebra provides a way to evaluate whether a consistent set of equations has an infinite number of solutions: a consistent system of n equations in n unknowns has a unique solution if the rank of the coefficient matrix is equal to its order, where r(A) = n. In the example h ...
... Another theorem of linear algebra provides a way to evaluate whether a consistent set of equations has an infinite number of solutions: a consistent system of n equations in n unknowns has a unique solution if the rank of the coefficient matrix is equal to its order, where r(A) = n. In the example h ...
Solution Key
... Find two elementary matrices E1 and E2 of type III so that E2 E1 A = I. Solution: We want to analyze which row operations of type III applied to A will yield I. We see that if we replace Row 3 with its sum with −2 times Row 2, and then replace Row 1 with its sum with 3 times Row 3, we obtain the ide ...
... Find two elementary matrices E1 and E2 of type III so that E2 E1 A = I. Solution: We want to analyze which row operations of type III applied to A will yield I. We see that if we replace Row 3 with its sum with −2 times Row 2, and then replace Row 1 with its sum with 3 times Row 3, we obtain the ide ...
Chapter 8: Matrices and Determinants
... A matrix gives us a shortened way of writing a system of equations. The first step in solving a system of linear equations using matrices is to write the augmented matrix. An augmented matrix has a vertical bar separating the columns of the matrix into tow groups. The coefficients of each variable a ...
... A matrix gives us a shortened way of writing a system of equations. The first step in solving a system of linear equations using matrices is to write the augmented matrix. An augmented matrix has a vertical bar separating the columns of the matrix into tow groups. The coefficients of each variable a ...
Solutions to Assignment 3
... 2.1.24 Suppose AD = Im (the m × m identity matrix). Show for any b in Rm , the equation Ax = b has a solution. Explain why A cannot have more columns then rows. Let b be any vector in Rm . Then Db is a vector as well, and A(Db) = (AD)b = Im b = b, so Ax = b always has a solution. By Theorem 4, page ...
... 2.1.24 Suppose AD = Im (the m × m identity matrix). Show for any b in Rm , the equation Ax = b has a solution. Explain why A cannot have more columns then rows. Let b be any vector in Rm . Then Db is a vector as well, and A(Db) = (AD)b = Im b = b, so Ax = b always has a solution. By Theorem 4, page ...
