# Questions about linear independent and spanning set and basi

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1. Jan 5, 2016

### DavidDai

• Member warned about posting with no effort shown
1. The problem statement, all variables and given/known data

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 R5

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!

Last edited: Jan 5, 2016
2. Jan 5, 2016

### Samy_A

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.

3. Jan 5, 2016

### DavidDai

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

4. Jan 5, 2016

### Samy_A

Thanks.

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

5. Jan 5, 2016

### DavidDai

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

6. Jan 5, 2016

### Samy_A

Using that third point, can you answer some of the questions?

7. Jan 5, 2016

### DavidDai

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

8. Jan 5, 2016

### Samy_A

Ok. What about questions 4 and 5?

9. Jan 5, 2016

### DavidDai

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

10. Jan 5, 2016

### Samy_A

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.

11. Jan 5, 2016

### DavidDai

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

12. Jan 5, 2016

### Staff: Mentor

Noted...

13. Jan 7, 2016

### DavidDai

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