text: # Math 3130 - Intro to Linear Algebra - MathOnline -...
  
  ```Math 3130 - Intro to Linear Algebra - MathOnline - Spring 2016
  Email: [email protected], Fax: 1-888-951-4321 (toll free)
  To be read and completed by the Proctor: By signing below, I certify that the exam was administered in the time allowed (75 minutes) and observing the following rules: no
  books/notes; scientific calculator (WITHOUT graphic or matrix capabilities) is allowed.
  Proctor Name:
  Proctor Signature (required)
  (Please fax or email the completed exam no later than Wed, April 20, 5pm MT.)
  Student Name:
  Exam 2 - Math 3130 - Intro to Linear Algebra
  The exam has 8 problems on 5 pages. Make sure you turn in ALL pages. To receive credit, show all work.
  No calculators with linear algebra capabilities allowed!
  Exercise 1.[3 pts each]
  Determine whether each of the following statements are TRUE or FALSE.
  Circle your answer and justify it for credit. No justification, no credit.
  • The set of nonzero row vectors of a matrix A is a basis for the row space of A
  TRUE
  FALSE
  • The reduced row echelon form of AT is the transpose of the reduced row echelon form of A.
  TRUE
  FALSE
  • Every linearly independent subset {v1 , . . . , vn } of a n-dimensional vector space is a basis.
  TRUE
  FALSE
  • The span of any finite set of vectors in a vector space V is a subspace of V .
  TRUE
  FALSE
  Exercise 2.[5 pts each]
  Decide whether Cramer’s rule can be used in solving each of the linear system below or not. If yes, solve it using
  Cramer’s rule. If not, explain why.
  (
  3x1 2x2 = 1
  (a)
  x1 2x2 = 2
  8
  &gt;
  &lt; x + 2y + 3z
  (b) x y + z
  &gt;
  :
  x + 5y + 5z
  =1
  =2
  =0
  Exercise 3.[5 pts each]
  (a) Find the complete solution of the linear system
  2
  1 2
  A = 42 6
  0 2
  Ax = b, where
  3
  2 3
  1
  2
  35 ,
  and b = 445
  1
  0
  (b) What is the rank of the matrix A? What is the column space of A?
  Exercise 4.[5 pts each]
  (a) Find a basis for the subspace of R3 spanned by the following 4 vectors
  2 3
  2 3
  2 3
  2 3
  1
  0
  2
  1
  v1 = 4 35 , v2 = 415 , v3 = 4 7 5 , v4 = 405
  0
  2
  2
  1
  (b) Express all the remaining vectors as linear combinations of the basis found in (a).
  Exercise 5.[9 pts]
  Find a basis for the subspace of R4 consisting of all vectors that are orthogonal to
  v1 = (1, 1, 2, 1)T and v2 = (0, 0, 1, 3)T
  .
  Exercise 6.[6 pts each]
  Given the matrix
  (1) Determine the rank of A.
  2
  1
  A = 41
  0
  (2) Find a basis for the column space of A.
  (3) Find a basis for the row space of A.
  (4) Find a basis for the nullspace space of A.
  2
  2
  0
  1
  0
  1
  4
  5
  1
  3
  1
  05
  1
  Exercise 7.[5 pts each]
  Find a basis and dimension of the following subspaces of R4
  (a) All vectors of the form (a, b, c, d) with a + b + c + d = 0
  (b) All vectors of the form (a, b, c, d) with a = b = c = d.
  Exercise 8.[5 pts each]
  Determine which of the following are subspaces of Mn⇥n , the space of all n ⇥ n matrices and which are not. Find
  the dimension of these subspaces.
  (a) The set of all diagonal n ⇥ n matrices.
  (b) The set of all n ⇥ n matrices A such that det(A) = 0.
  (c) The set of all symmetric matrices A (AT = A)
  ```
explanation: The document is an exam for a course titled "Math 3130 - Intro to Linear Algebra." It contains a series of exercises and problems that students are required to solve. Each exercise is a math question, and the document does not provide answers to these questions.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Math 3130 - Intro to Linear Algebra - MathOnline -...
  
  ```Math 3130 - Intro to Linear Algebra - MathOnline - Spring 2016
  Email: [email protected], Fax: 1-888-951-4321 (toll free)
  To be read and completed by the Proctor: By signing below, I certify that the exam was administered in the time allowed (75 minutes) and observing the following rules: no
  books/notes; scientific calculator (WITHOUT graphic or matrix capabilities) is allowed.
  Proctor Name:
  Proctor Signature (required)
  (Please fax or email the completed exam no later than Wed, April 20, 5pm MT.)
  Student Name:
  Exam 2 - Math 3130 - Intro to Linear Algebra
  The exam has 8 problems on 5 pages. Make sure you turn in ALL pages. To receive credit, show all work.
  No calculators with linear algebra capabilities allowed!
  Exercise 1.[3 pts each]
  Determine whether each of the following statements are TRUE or FALSE.
  Circle your answer and justify it for credit. No justification, no credit.
  • The set of nonzero row vectors of a matrix A is a basis for the row space of A
  TRUE
  FALSE
  • The reduced row echelon form of AT is the transpose of the reduced row echelon form of A.
  TRUE
  FALSE
  • Every linearly independent subset {v1 , . . . , vn } of a n-dimensional vector space is a basis.
  TRUE
  FALSE
  • The span of any finite set of vectors in a vector space V is a subspace of V .
  TRUE
  FALSE
  Exercise 2.[5 pts each]
  Decide whether Cramer’s rule can be used in solving each of the linear system below or not. If yes, solve it using
  Cramer’s rule. If not, explain why.
  (
  3x1 2x2 = 1
  (a)
  x1 2x2 = 2
  8
  &gt;
  &lt; x + 2y + 3z
  (b) x y + z
  &gt;
  :
  x + 5y + 5z
  =1
  =2
  =0
  Exercise 3.[5 pts each]
  (a) Find the complete solution of the linear system
  2
  1 2
  A = 42 6
  0 2
  Ax = b, where
  3
  2 3
  1
  2
  35 ,
  and b = 445
  1
  0
  (b) What is the rank of the matrix A? What is the column space of A?
  Exercise 4.[5 pts each]
  (a) Find a basis for the subspace of R3 spanned by the following 4 vectors
  2 3
  2 3
  2 3
  2 3
  1
  0
  2
  1
  v1 = 4 35 , v2 = 415 , v3 = 4 7 5 , v4 = 405
  0
  2
  2
  1
  (b) Express all the remaining vectors as linear combinations of the basis found in (a).
  Exercise 5.[9 pts]
  Find a basis for the subspace of R4 consisting of all vectors that are orthogonal to
  v1 = (1, 1, 2, 1)T and v2 = (0, 0, 1, 3)T
  .
  Exercise 6.[6 pts each]
  Given the matrix
  (1) Determine the rank of A.
  2
  1
  A = 41
  0
  (2) Find a basis for the column space of A.
  (3) Find a basis for the row space of A.
  (4) Find a basis for the nullspace space of A.
  2
  2
  0
  1
  0
  1
  4
  5
  1
  3
  1
  05
  1
  Exercise 7.[5 pts each]
  Find a basis and dimension of the following subspaces of R4
  (a) All vectors of the form (a, b, c, d) with a + b + c + d = 0
  (b) All vectors of the form (a, b, c, d) with a = b = c = d.
  Exercise 8.[5 pts each]
  Determine which of the following are subspaces of Mn⇥n , the space of all n ⇥ n matrices and which are not. Find
  the dimension of these subspaces.
  (a) The set of all diagonal n ⇥ n matrices.
  (b) The set of all n ⇥ n matrices A such that det(A) = 0.
  (c) The set of all symmetric matrices A (AT = A)
  ```
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an exam for a course titled "Math 3130 - Intro to Linear Algebra." It contains a series of exercises and problems that students are required to solve. Each exercise is a math question, and the document does not provide answers to these questions.
  
  Label: multiple-question