Turn on thread page Beta
    • Thread Starter
    Offline

    1
    ReputationRep:
    Hi, I'm struggling to tell whether the following are subspaces.
    If I'm correct, I must show that the zero vector is in the subspace, they're closed under addition and closed under multiplication. It's just I'm not sure how to do this.

    For the first one, can we let x1,2,3,4=0 and x5=-1 to show the zero vector exits and then for the second let x1,2,3=0 to show the zero vector exits.

    I'm struggling with the next step, Thanks!

    Name:  subspaces.png
Views: 42
Size:  10.5 KB
    Offline

    22
    ReputationRep:
    (Original post by Substitution)
    Hi, I'm struggling to tell whether the following are subspaces.
    If I'm correct, I must show that the zero vector is in the subspace, they're closed under addition and closed under multiplication. It's just I'm not sure how to do this.

    For the first one, can we let x1,2,3,4=0 and x5=-1 to show the zero vector exits and then for the second let x1,2,3=0 to show the zero vector exits.

    I'm struggling with the next step, Thanks!
    Following what? You haven't posted the question. :-)
    Offline

    13
    ReputationRep:
    (Original post by Substitution)
    Hi, I'm struggling to tell whether the following are subspaces.
    If I'm correct, I must show that the zero vector is in the subspace, they're closed under addition and closed under multiplication. It's just I'm not sure how to do this.

    For the first one, can we let x1,2,3,4=0 and x5=-1 to show the zero vector exits and then for the second let x1,2,3=0 to show the zero vector exits.

    I'm struggling with the next step, Thanks!
    Can you be a little clearer with your question, it's not at all obvious what you're saying.
    You're right about the conditions, though it's worth saying that the multiplication is scalar multiplication, and you can combine these conditions to say that if \mathbf{u}, \mathbf{v} \in V where V is the subspace, then we require \lambda \mathbf{u}+\mu \mathbf{v} \in V.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by Zacken)
    Following what? You haven't posted the question. :-)
    (Original post by joostan)
    Can you be a little clearer with your question, it's not at all obvious what you're saying.
    You're right about the conditions, though it's worth saying that the multiplication is scalar multiplication, and you can combine these conditions to say that if \mathbf{u}, \mathbf{v} \in V where V is the subspace, then we require \lambda \mathbf{u}+\mu \mathbf{v} \in V.
    Sorry! Just edited my first post to include the question
    Offline

    13
    ReputationRep:
    (Original post by Substitution)
    Sorry! Just edited my first post to include the question
    Well for the second, you're correct, but for the first, the zero vector is clearly not in U, further setting x_5=-1 and the others\ 0 doesn't mean that this is the zero vector, and further this vector is not in U.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by joostan)
    Well for the second, you're correct, but for the first, the zero vector is clearly not in U, further setting x_5=-1 and the others\ 0 doesn't mean that this is the zero vector, and further this vector is not in U.
    Oh yes of course, don't know how I cold be so oblivious to that. My bad! Would you be able to direct me to how show its clsoed under addition (for practice) Thanks
    Offline

    13
    ReputationRep:
    (Original post by Substitution)
    Oh yes of course, don't know how I cold be so oblivious to that. My bad! Would you be able to direct me to how show its clsoed under addition (for practice) Thanks
    Write \mathbf{x}=(x_1,x_2,x_3)^T , \ \mathbf{y}=(y_1,y_2,y_3)^T and consider the sum \lambda \mathbf{x}+ \mu \mathbf{y} for \lambda, \mu \in \mathbb{R}, you'll need to use the property 3x_1-x_2-2x_3=3y_1-y_2-2y_3=0.
    If this is in U then this simultaneously shows that U is closed under addition and scalar multiplication.

    Alternatively you can check these separately, by considering \mathbf{x}+\mathbf{y} and \lambda \mathbf{x}.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by joostan)
    Write \mathbf{x}=(x_1,x_2,x_3)^T , \ \mathbf{y}=(y_1,y_2,y_3)^T and consider the sum \lambda \mathbf{x}+ \mu \mathbf{y} for \lambda, \mu \in \mathbb{R}, you'll need to use the property 3x_1-x_2-2x_3=3y_1-y_2-2y_3=0.
    If this is in U then this simultaneously shows that U is closed under addition and scalar multiplication.

    Alternatively you can check these separately, by considering \mathbf{x}+\mathbf{y} and \lambda \mathbf{x}.
    Hi, Thanks for explaining that.

    Still not sure about knowing whether it is in U. Could you please explain a little further
    Offline

    13
    ReputationRep:
    (Original post by Substitution)
    Hi, Thanks for explaining that.

    Still not sure about knowing whether it is in U. Could you please explain a little further
    Well if \mathbf{v}=\mathbf{x}+\mathbf{y} then \mathbf{v} \in U \iff 3v_1-v_2-2v_3=0.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: March 19, 2016
The home of Results and Clearing

3,010

people online now

1,567,000

students helped last year

University open days

  1. Bournemouth University
    Clearing Open Day Undergraduate
    Wed, 22 Aug '18
  2. University of Buckingham
    Postgraduate Open Evening Postgraduate
    Thu, 23 Aug '18
  3. University of Glasgow
    All Subjects Undergraduate
    Tue, 28 Aug '18
Poll
How are you feeling about GCSE results day?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.