Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Subspaces - Linear Algebra - Exam, Exams of Linear Algebra

This is the Past Exam of Linear Algebra which includes Vectors, Angle, System, Factorization, Matrix Pascal, Matrix, Reflects Vectors, Line Making Angle, Same Line etc. Key important points are: Subspaces Associated, Complete Solution, System, Factorization, Determinant, Solve, Basis, Associated, Factored, Column Space

Typology: Exams

2012/2013

Uploaded on 02/27/2013

sehgal_98
sehgal_98 šŸ‡®šŸ‡³

4.8

(4)

137 documents

1 / 1

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Exam #2, Math 205B (Linear Algebra)
This take-home exam is due on Friday, November 14 by 5 PM. You may consult the textbook (or any
other book) and any class notes and handouts, but please do not discuss any details of this exam
with anyone except me! Please sign the bottom of this sheet and turn it in with your exam. You may ask
me questions about the exam, but I reserve the right to give unsatisfying answers. Matrix multiplications
and reduced row echelon forms may be done on MATLAB or a calculator, but please show all other work.
1. (16 points) Find the complete solution of the system 

13122
12143
14221







x1
x2
x3
x4
x5





=

3
4
2

.
2. (20 points) Find a basis for each of the four subspaces associated with the matrix
A=


1 1 3 5
2 1 2 7
2 3 10 13
5 3 7 19



What is the factored form of Athat displays these bases?
3. (14 points) The vectors 


1
1
1
2


,


1
2
1
5


and 


1
4
1
4


span a 3-dimensional subspace of R4. Find an orthonormal
basis for it.
4. (18 points) Let ~v1=


1
3
1
1


,~v2=


1
1
3
1


, and ~v3=


3
3
1
āˆ’1


, and let Sbe the subspace of R4spanned by
~v1and ~v2. Find the matrix Pthat projects vectors in R4onto S, and the matrix Rthat reflects through S.
Find also the projection ~p of ~v3onto Sand the reflection ~r of ~v3through S.
5. (18 points) Explain how you can tell that P=1
12







5 2 āˆ’1āˆ’1 2 5
2 2 2 2 2 2
āˆ’1 2 5 5 2 āˆ’1
āˆ’1 2 5 5 2 āˆ’1
2 2 2 2 2 2
5 2 āˆ’1āˆ’1 2 5







is a projection matrix.
Find a basis for the subspace Tof R6that Pprojects onto, and a basis for T⊄(the orthogonal complement
of T).
6. (8 points) Continuing a theme from the third homework set, construct a 4 Ɨ4 matrix whose column space
equals its nullspace.
I affirm that I did not receive help from another person in doing this exam, nor did I give help
to another student in the class.
(signed)

Partial preview of the text

Download Subspaces - Linear Algebra - Exam and more Exams Linear Algebra in PDF only on Docsity!

Exam #2, Math 205B (Linear Algebra)

This take-home exam is due on Friday, November 14 by 5 PM. You may consult the textbook (or any other book) and any class notes and handouts, but please do not discuss any details of this exam with anyone except me! Please sign the bottom of this sheet and turn it in with your exam. You may ask me questions about the exam, but I reserve the right to give unsatisfying answers. Matrix multiplications and reduced row echelon forms may be done on MATLAB or a calculator, but please show all other work.

  1. (16 points) Find the complete solution of the system

x 1 x 2 x 3 x 4 x 5

  1. (20 points) Find a basis for each of the four subspaces associated with the matrix

A =

What is the factored form of A that displays these bases?

  1. (14 points) The vectors

 and

 span a 3-dimensional subspace of R^4. Find an orthonormal

basis for it.

  1. (18 points) Let ~v 1 =

,^ ~v 2 =

, and^ ~v 3 =

, and let^ S^ be the subspace of^ R^4 spanned by

~v 1 and ~v 2. Find the matrix P that projects vectors in R^4 onto S, and the matrix R that reflects through S. Find also the projection ~p of ~v 3 onto S and the reflection ~r of ~v 3 through S.

  1. (18 points) Explain how you can tell that P =

is a projection matrix.

Find a basis for the subspace T of R^6 that P projects onto, and a basis for T ⊄^ (the orthogonal complement of T ).

  1. (8 points) Continuing a theme from the third homework set, construct a 4 Ɨ 4 matrix whose column space equals its nullspace.

I affirm that I did not receive help from another person in doing this exam, nor did I give help to another student in the class.

(signed)