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In summary: You can also think about it geometrically: each of the three vectors represents a different direction in 3D space, and together they can generate any point in R3.
- #2
Mark44
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judahs_lion said:
Not right. In a nutshell you want to show that for an arbitrary vector <x, y, z>, there are some constants a, b, and c so that aV1 +bV2 +cV3 = <x,y,z>.
You can do this by solving the matrix equation Ab = x for b, where the columns of matrix A are your vectors V1, V2, and V3. The vector I show as b is <a, b, c>, and the vector I show as x is <x, y, z>.
- #3
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Try showing that you can generate the standard basis of R3, {(1,0,0), (0,1,0), (0,0,1)}, using the elements v1, v2, v3. For example, what combination of these vectors will give you (0,1,0)?
- #4
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Is there any way u dumb it down just a lil more , I still feel very lost.
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- #6
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You want to show that {v1, v2, v3} = V spans R3. You already know that the vectors (1,0,0), (0,1,0), and (0,0,1) span R3. So you can try showing that V generates (0,1,0) and (0,0,1) and thus generates R3
- #7
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How do I know they span R3?
- #8
Mark44
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judahs_lion said:
The work in the attachment looks fine. If you can row-reduce your matrix to the identity matrix [1 0 0; 0 1 0; 0 0 1], that's enough to guarantee that your three vectors span R3.
1. What does it mean for vectors to span R3?
When we say that vectors span R3, it means that these vectors are able to create any point in the 3-dimensional space of R3 through linear combinations. In other words, any vector in R3 can be represented as a linear combination of these spanning vectors.
2. How can we prove that vectors span R3?
To prove that vectors span R3, we need to show that any vector in R3 can be written as a linear combination of the spanning vectors. This can be done through Gaussian elimination or by solving a system of linear equations.
3. Can a set of two vectors span R3?
No, a set of two vectors cannot span R3 because R3 is a 3-dimensional space and it requires at least three linearly independent vectors to span it. Two vectors can only span a 2-dimensional subspace of R3.
4. What is the minimum number of vectors needed to span R3?
The minimum number of vectors needed to span R3 is three. This is because R3 is a 3-dimensional space and it requires three linearly independent vectors to create any point in it through linear combinations.
5. How are vectors tested for linear independence when proving they span R3?
When proving that vectors span R3, we also need to show that these vectors are linearly independent. This can be done by checking if the determinant of the matrix formed by these vectors is non-zero, or by using the concept of rank to determine if the vectors are linearly independent.