Proving something to be a basis.

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In summary, the conversation discusses how to show that a general vector can be written as a linear combination of a given basis without using an augmented matrix. Various methods are suggested, including using the linear independence of the basis vectors and considering the dimension of their span. Another example is given with polynomials, where it is shown that a polynomial with degree higher than 2 cannot be written as a linear combination of four polynomials with degree <= 2.
  • #1
dylanhouse
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Homework Statement



Letting u=[3, 0, -5], v=[2, 1, 5] and w=[-1, 3, 4], how would I show that a general vector can be written as a linear combination of this 'basis?' Without using an augmented matrix and getting a really messy result by using arbitrary a, b, and c values as the solutions?

Homework Equations





The Attempt at a Solution

 
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  • #2
Can you show that the three elements of the canonical basis, namely [1, 0, 0], [0, 1, 0], and [0, 0, 1], can be written as linear combinations of the given vectors? If so, can you see how that helps?
 
  • #3
dylanhouse said:

Homework Statement



Letting u=[3, 0, -5], v=[2, 1, 5] and w=[-1, 3, 4], how would I show that a general vector can be written as a linear combination of this 'basis?' Without using an augmented matrix and getting a really messy result by using arbitrary a, b, and c values as the solutions?

Homework Equations





The Attempt at a Solution


So much depends on what you know and are allowed to use. For example, do you know the connection between linear independence and determinants? If so, you can use that.

Otherwise, you can try to show directly that u, v and w are linearly independent, so form a basis; then you can quote a theorem to finish the problem. Or, do you really want to express an arbitrary vector x = [x1,x2,x3] as an explicit linear combination of u, v and w? The problem statement seemed to say otherwise, but only you know for sure.
 
  • #4
I have shown that they are linearly independent. But I also need to show that the Span(s)=V.
 
  • #5
If you have three linearly independent vectors, what can you say about the dimension of their span?
 
  • #6
I figured it out :) I was also wondering if I am given four polynomials, of which the highest degree is x^2, they cannot be a basis for the vector space P3(R)?
 
  • #7
What is P3(R)? The space of polynomials with degree <= 3? If so, consider whether it's possible to write ##x^3## as a linear combination of the four given polynomials.
 
  • #8
Well, it wouldn't be possible unless my scalar was the variable x, and not a real number.
 
  • #9
Can you prove why it wouldn't be possible?
 
  • #10
Well, I converted the polynomials to a matrix, and tried to put it into RREF to prove they were linearly independent. But the first column is all 0's, so it was pointless to go any further as they were obviously linearly dependent and thus not a basis I assumed.
 
  • #11
OK, that's reasonable but you can also make an argument without even knowing anything about the four polynomials ##p_1(x),\ldots,p_4(x)## except that none of their degrees are higher than 2.

Then suppose ##x^3## can be written as a linear combination of those four polynomials. We would then have
$$x^3 + \sum_{n=1}^{4}a_n p_n(x) = 0$$
for some scalars ##a_1,\ldots,a_4##. But ##x^3 + \sum_{n=1}^{4}a_n p_n(x)## is a polynomial of degree 3 (why?), so it cannot equal the zero polynomial (why?)
 

1. What is a basis in mathematics?

A basis in mathematics refers to a set of linearly independent vectors that can be used to express any vector in a given vector space. It is the smallest set of vectors that can span the entire vector space.

2. What does it mean to prove something to be a basis?

Proving something to be a basis means showing that the set of vectors in question satisfies the definition of a basis, i.e. they are linearly independent and span the entire vector space.

3. How do you prove that a set of vectors is linearly independent?

To prove that a set of vectors is linearly independent, you can use the definition of linear independence, which states that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where ci are constants and vi are the vectors in the set, is when all the constants are equal to 0.

4. How do you show that a set of vectors spans a vector space?

To show that a set of vectors spans a vector space, you can prove that any vector in the space can be written as a linear combination of the vectors in the set. This means that for any vector v in the space, there are constants ci such that v = c1v1 + c2v2 + ... + cnvn.

5. Can a set of vectors be both linearly independent and spanning?

Yes, a set of vectors can be both linearly independent and spanning, which means it is a basis for the vector space. This is because a basis satisfies both requirements: linear independence and spanning the entire space.

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