Linear algebra, subspace of a vector space?

In summary, the conversation discusses whether a set of polynomials of the form p(x) = a + x^3, with a in R, is a subspace of a vector space P sub6 of all polynomials of degree at most 6. The properties of a subspace are also mentioned, including the zero vector, closure under vector addition, and closure under scalar multiplication. The conclusion is that the set is not a subspace because it does not satisfy all three properties.
  • #1
toyotadude
18
0

Homework Statement



1) The set H of all polynomials p(x) = a+x^3, with a in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False?

2) The set H of all polynomials p(x) = a+bx^3, with a,b in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False?

Homework Equations



Eh.. not sure?

The Attempt at a Solution



Once more, not too sure. I've been pouring over my Linear Algebra book, but it seems so abstract... I was under the impression that as long as the polynomial didn't have a lower power than the vector space [number] that the polynomial would be in the subspace of the given vector space :\

Does the coefficient have anything to do with it?

Some properties (if they help?): A subspace of a vector space V is a subset H of V that has 3 properties:
a) The zero vector if V is in H.
b) H closed under vector addition
3) H closed under scalar multiplication..

I've already gotten #1 wrong (the answer was false) - I'd like to know why though :(

Any help would be awesome!
 
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  • #2
toyotadude said:

Homework Statement



1) The set H of all polynomials p(x) = a+x^3, with a in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False?
Some properties (if they help?): A subspace of a vector space V is a subset H of V that has 3 properties:
a) The zero vector if V is in H.
b) H closed under vector addition
3) H closed under scalar multiplication..

I've already gotten #1 wrong (the answer was false) - I'd like to know why though :(
Consider property (a). Can the zero vector (polynomial in this case) be written in the form ##p(x) = a + x^3##?

Also consider property (b). If I have two polynomials of the form ##a + x^3##, and add them together, is the result also of the form ##a + x^3##?

Finally, consider property (c). If I multiply a polynomial of the form ##a + x^3## by an arbitrary scalar, say ##2##, is the result of the form ##a + x^3##?
 
  • #3
toyotadude said:
Once more, not too sure. I've been pouring over my Linear Algebra book, but it seems so abstract.

What exactly were you pouring over your book? Coffee? Beer?? And why? :confused:
 
  • #4
LCKurtz said:
What exactly were you pouring over your book? Coffee? Beer?? And why? :confused:
Yeah, not a good idea - the pages will stick together. :tongue:
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vector spaces, linear transformations, and systems of linear equations. It involves the use of algebraic structures such as matrices and determinants to solve problems related to these concepts.

2. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and a set of operations (such as addition and scalar multiplication) that can be performed on those vectors. These operations must follow certain rules, such as closure under addition and scalar multiplication, to qualify as a vector space.

3. What is a subspace?

A subspace is a subset of a vector space that also satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector. Subspaces are useful in linear algebra because they allow us to study smaller, more manageable parts of a larger vector space.

4. How do you determine if a set of vectors forms a subspace?

To determine if a set of vectors forms a subspace, we must check if it satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and contains the zero vector. This can be done by performing the necessary operations on the vectors and checking if the results are still within the original set.

5. What are some applications of linear algebra in real life?

Linear algebra has numerous applications in various fields such as physics, engineering, computer science, and economics. It is used to solve systems of equations, analyze data, and create mathematical models for real-world problems. Some specific applications include image and signal processing, machine learning, and optimization.

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