Subset & Subspace Homework: Closed Under Vector Addition & Scalar Multiplication

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In summary: You need to find one set for each.In summary, the student is trying to find two different sets of vectors in R2, one that is closed under vector addition but not scalar multiplication and one that is closed under scalar multiplication but not vector addition. They have attempted a solution using a set S = {(x,y) | x + y = 0} but this is incorrect as it is a one dimensional subspace of R2. They are asked to look at examples in their notes or textbook to find the correct sets.
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Homework Statement




a) Find a set of vectors in R2 that is closed under vector addition but not under scalar multiplication
Find a set of vectors closed under scalar multiplication but not closed under vector addition.

The Attempt at a Solution



a) Let S be a set of vectors in R2.

S = {(x,y) | x + y =0}
x = (1,1) y = (-1,-1)

To show that S set of vectors is closed under vector addition, x + y must remain in S.

x + y = (x1 + y1, x2+y2) = ( 0,0)

Am I right up till here?
 
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negation said:

Homework Statement

a) Find a set of vectors in R2 that is closed under vector addition but not under scalar multiplication
Find a set of vectors closed under scalar multiplication but not closed under vector addition.

The Attempt at a Solution



a) Let S be a set of vectors in R2.

S = {(x,y) | x + y =0}
x = (1,1) y = (-1,-1)

To show that S set of vectors is closed under vector addition, x + y must remain in S.

x + y = (x1 + y1, x2+y2) = ( 0,0)

Am I right up till here?
No. Your set S is the line whose equation is x + y = 0. This is a line through the origin, and as such, this set is a one dimensional subspace of R2. You need to find a different set of vectors.

Look at the examples in your book or notes. There are probably some examples of sets that are closed under one operation, but not the other.

Also note that this is two problems - one for a set that is closed under vector addition but not under multiplication by a scalar; the other is closed under scalar multiplication but not vector addition.
 

FAQ: Subset & Subspace Homework: Closed Under Vector Addition & Scalar Multiplication

What is a subset?

A subset is a set that contains elements which are all also elements of another set. In other words, all the elements in a subset must also be found in the original set.

What does it mean for a subset to be closed under vector addition?

A subset is closed under vector addition if when two vectors from the subset are added together, the result is still a vector in the subset. In other words, the subset contains all possible combinations of vectors that can be formed by adding vectors from the subset together.

What does it mean for a subset to be closed under scalar multiplication?

A subset is closed under scalar multiplication if when a vector from the subset is multiplied by a scalar, the result is still a vector in the subset. In other words, the subset contains all possible scalar multiples of the vectors within the subset.

Why is it important for a subset to be closed under vector addition and scalar multiplication?

Closedness under vector addition and scalar multiplication is important because it ensures that the subset is a vector space, meaning it follows all the necessary properties and operations of a vector space. This is useful in mathematical and scientific applications where vector operations are often used.

How do I determine if a subset is closed under vector addition and scalar multiplication?

To determine if a subset is closed under vector addition, you can add any two vectors from the subset together and see if the result is still in the subset. To determine if a subset is closed under scalar multiplication, you can multiply any vector from the subset by any scalar and see if the result is still in the subset. If both of these conditions are true, then the subset is closed under both vector addition and scalar multiplication.

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