Linear Algebra Midterm Exam 1
Name:
Time: 11am-12:50am, 10/7/2015
Problem
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7
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Points
Problem 1 (10 points)
1
Let v = 2.
3
(1) Find the length of v.
(2) Find two independent vectors w
Written Homework 6 (57 points)
Math for Economics I Summer Session 1, 2015
Due Monday, June 29, by the start of class
Write neatly Show all your work to get full credit Write your name Staple multiple
Name:
QUIZ 7: Vector spaces and coordinate vectors.
The set of vectors B = 1, 1 t, t2 4t + 2 forms a basis for P2 , the vector space of polynomials
of degree 2.
1.) Find [x]B , the coordinate vector o
Linear Algebra I ExamSummer 2011Solutions
1. Let u1 = (2, 3, 1), u2 = (1, 4, 2), u3 = (8, 12, 4), u4 =
(1, 37, 17), u5 = (3, 5, 8). Find a basis for R3 consisting of vectors
uk .
Solution 1. It is not
2011 Linear Algebra I Mid-Term Exam
1. Which of the following are vector spaces?
(a) Let F be a eld and V = F2 . Let addition in V be the usual one (coordinatewise)
and scalar multiplication be dened
MATH-UA 140 (Linear Algebra) Homework 5 Solutions
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Problem 1 (5 parts): Consider the vector spaces Rnn for any n cfw_1, 2, . . ..
(a) Show
MATH-UA 140 (Linear Algebra) Homework 7
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Due: Nov. 10 (before midnight; see syllabus for submission rules)
Note: These problems are based
MATH-UA 140 (Linear Algebra) Homework 8
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Due: Nov. 22 (before midnight; see syllabus for submission rules)
Note: These problems are based
MATH-UA 140 (Linear Algebra) Homework 6 Solutions
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Problem 1 (7 parts):
(a) Compute the dimensions of the fundamental subspaces of any ran
MATH-UA 140 (Linear Algebra) Homework 2
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Due: Sept. 17 (before midnight; see syllabus for submission rules)
Note: These problems are based
MATH-UA 140 (Linear Algebra) Homework 4 Solutions
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Problems 1 through 5 study properties of vector spaces. A vector space is a set V that
MATH-UA 140 (Linear Algebra) Homework 1 Solutions
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Problem 1: If three corners of a parallelogram are (1, 0), (0, 1), and (1, 0), what are
MATH-UA 140 (Linear Algebra) Homework 3 Solutions
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Problem 1 (5 parts): Write down the 3-by-3 matrices corresponding to the following thre
MATH-UA 140 (Linear Algebra) Homework 3
Section: 002, Semester: Fall 2015, Instructor: Nicholas Knight
Due: Sept. 26 (before midnight; see syllabus for submission rules)
Note: These problems are based
Written Homework 2 (44 points)
Math for Economics I Summer Session 1, 2015
Due Monday, June 8, by the start of class
Write neatly Show all your work to get full credit Write your name Staple multiple
Lecture 1: Vectors
1.1 Vectors and Linear Combination
Denition of a vector, notations
Vector addition and scalar multiplication
Linear combination of vectors
Visualization in 2-d or 3-d spaces
1.2
Lecture 10: The Complete Solution to Ax = b
3.4 The Complete Solution to Ax = b
The conditions that Ax = b has solutions.
The particular solution xp : the solution to Ax = b such that all free varia
Lecture 18: Applications of Determinants
5.3 Cramers Rule, Inverse, and Volume
Cramers Rule If det A = 0, Ax = b is solved by determinants
xj =
det Bj
det A
(1)
where the matrix Bj has the j-th of A
Lecture 16: Determinant
4.4 Gram-Schmidt (continued)
The Gram-Schmidt process creates orthonormal vectors from independent vectors. Suppose
we have n independent vectors v1 , v2 , , vn , we generate
Lecture 12: The Four Subspaces
3.6 Dimensionans of the Four Subspaces
The four fundamental subspaces for an m by n matrix A: the row space C(AT ), a subspace
of Rn ; the column space C(A), a subspace
Lecture 17: Formulas for Determinant
5.2 Permutations and Cofactors
det A = (d1 d2 dn ) where the sign depends on the parity of how many row exchanges
in the elimination, and d1 , d2 , , dn are the p
Lecture 13: Projections
4.2 Projections
Projection onto a line: projecting b onto a, the projection is p = xa =
the error is e = b xa.
T
The projection matrix: P = aa a . Note that P 2 = P , (I P )2
Lecture 15: Orthogonal Bases and Gram-Schmidt
4.4 Orthogonal Bases and Gram-Schmidt
The vectors q1 , , qn are orthonormal if
T
qi qj =
0, when i = j
1, when i = j
(orthogonal vectors)
(unit vectors:
Lecture 14: Least Square Approximations
4.2 Projections(continued)
Properties of projection matrix P : P 2 = P , (I P )2 = I P .
The projection of v onto a subspace W is the closest vector to v in W
Lecture 4: Matrix Multiplication
2.3-2.4 Elimination Using Matrices and Matrix Multiplication
The matrix multiplication AB is only dened when
number of columns of A = number of rows of B
What is the
Lecture 5: Inverse Matrices
(We only consider square matrix in this section)
2.3-4 Elimination Using Matrices and Matrix Multiplication
Sovling Ax = b by using Eij , Pij and the augmented matrix.
2.5
Lecture 6: LU Factorization and Transposes
2.6 LU Factorization
The elimination without row exchanges can be expressed as A = LU , where U is an upper
triangular matrix with pivots on its diagonal, a
Lecture 8: Null Space
3.2 The Nullspace of A: Solving Ax = 0
Denition: The null space of A consists of all solutions to Ax = 0. The notation is N (A).
Special solutions: the solution to Ax = 0 obtai