Questions about linear independent and spanning set and basi

[DavidDai]

Homework Statement

Definition of spanning set:
Let be vectors in the vector space . The set of all linear combination of the vectors is a subspace ( say ) of . The subspace is called the space spanned by the vectors The set is called a spanning set of .

Definition of linear independence:
Suppose that is a vector space. The set of vectors from is linearly dependent if there is a relation oflinear dependence on that is not trivial. In the case where the only relation of linear dependence on is the trivial one, then is a linearly independent set of vectors.

Definition of basis of vector space:
1. It spans the space.
2. Its vectors are independent.
3. The number of vectors in the basis is equal to the dimension of the space.
True or false? Given reason

1. A set of 5 Vectors in R5 must be a basis for R5
2. A set of 6 Vectors in R5 cannot be a basis for R5
3. A set of 7 vectors in R5 must be a spanning set for R5
4. A set of 6 polynomials in R5 must be a basis for R5
5. A set of 6 polynomials in R5 may be a basis for R5Anyone help me to explain these question.. I want to know the reason cz it's confused me a lot and i always get mess about these kind of question
appericated it!

 

[Samy_A]

Homework Statement

True or false? Given reason

Homework Equations

1. A set of 5 Vectors in R5 must be a basis for R5
2. A set of 6 Vectors in R5 cannot be a basis for R5
3. A set of 7 vectors in R5 must be a spanning set for R5
4. A set of 6 polynomials in R5 must be a basis for R5
5. A set of 6 polynomials in R5 may be a basis for R5

The Attempt at a Solution

Anyone help me to explain these question.. I want to know the reason cz it's confused me a lot and i always get mess about these kind of question
appericated it!

I assume that what you wrote under "Relevant equations" is actually the "The problem statement", and that R5 stand for $\mathbb R^5$.

For starters, maybe you could define the used concepts: basis and spanning set.

 

[DavidDai]

I assume that what you wrote under "Relevant equations" is actually the "The problem statement", and that R5 stand for $\mathbb R^5$.

For starters, maybe you could define the used concepts: basis and spanning set.

Thanks. I just edited the form of the question and now it should be ok..btw, Can u help me to this question?

 

[Samy_A]

Thanks. I just edited the form of the question and now it should be ok..btw, Can u help me to this question?

Thanks.

The forum rules expect you to show some effort in solving the exercise, before other forum members jump into help.
That's why I suggested you at least start with defining what a basis is, and what a spanning set is.

 

[DavidDai]

Thanks.

The forum rules expect you to show some effort in solving the exercise, before other forum members jump into help.
That's why I suggested you at least start with defining what a basis is, and what a spanning set is.

Thanks for reminding. This is my first time to ask question here so sorry about that.

 

[Samy_A]

Homework Statement

Definition of spanning set:
Let be vectors in the vector space . The set of all linear combination of the vectors is a subspace ( say ) of . The subspace is called the space spanned by the vectors The set is called a spanning set of .

Definition of linear independence:
Suppose that is a vector space. The set of vectors from is linearly dependent if there is a relation oflinear dependence on that is not trivial. In the case where the only relation of linear dependence on is the trivial one, then is a linearly independent set of vectors.

Definition of basis of vector space:
1. It spans the space.
2. Its vectors are independent.
3. The number of vectors in the basis is equal to the dimension of the space.

(I added the bold blue)
Using that third point, can you answer some of the questions?

 

[DavidDai]

(I added the bold blue)
Using that third point, can you answer some of the questions?

I can answer question 2 the reason is any set of 6 vectors in R5 is linearly dependent, even though some sets of 6 vectors in R5 span R5. To be a basis it must be a linearly independent spanning set, so if it's linearly dependent, it cannot be a basis.
But actually I know the answer for all of these questions but i don't the reason apart from question2

 

[Samy_A]

I can answer question 2 the reason is any set of 6 vectors in R5 is linearly dependent, even though some sets of 6 vectors in R5 span R5. To be a basis it must be a linearly independent spanning set, so if it's linearly dependent, it cannot be a basis.
But actually I know the answer for all of these questions but i don't the reason apart from question2

Ok. What about questions 4 and 5?

 

[DavidDai]

Ok. What about questions 4 and 5?

No i don't know how to explain question 1 3 4 5.

 

[Samy_A]

No i don't know how to explain question 1 3 4 5.

Let's take 4 as an example:
"4. A set of 6 polynomials in R5 must be a basis for R5"

Can a set of 6 polynomials be a basis for $\mathbb R^5$? The answer can be found using the definitions you posted.

 

[DavidDai]

Let's take 4 as an example:
"4. A set of 6 polynomials in R5 must be a basis for R5"

Can a set of 6 polynomials be a basis for $\mathbb R^5$? The answer can be found using the definitions you posted.

Thanks for helping. Here is too late now so I have to sleep. I am going to do these question tomorrow.
Appericated for helping!

 

[Mark44]

The forum rules expect you to show some effort in solving the exercise, before other forum members jump into help.

Noted...

 

[DavidDai]

Hi guys. I hv done these question already. Thanks for helping!