Name two sets of vectors that could be used to span a set in R^3. Show how the vectors (-1,2,0) and (3,4,0) could each be written as a linear combination of the vectors you have chosen.
Two sets of vectors that could be used to span a set in R^3 are:
Set 1: { (1, 0, 0), (0, 1, 0), (0, 0, 1) }
Set 2: { (1, 1, 0), (0, 1, 1), (1, 0, 1) }
Now, let's show how the vectors (-1, 2, 0) and (3, 4, 0) can be written as linear combinations of the vectors in each set:
For Set 1:
-1 = -1*(1, 0, 0) + 0*(0, 1, 0) + 0*(0, 0, 1)
2 = 0*(1, 0, 0) + 2*(0, 1, 0) + 0*(0, 0, 1)
So, (-1, 2, 0) can be written as a linear combination of the vectors in Set 1.
3 = 3*(1, 0, 0) + 0*(0, 1, 0) + 0*(0, 0, 1)
4 = 0*(1, 0, 0) + 4*(0, 1, 0) + 0*(0, 0, 1)
So, (3, 4, 0) can also be written as a linear combination of the vectors in Set 1.
For Set 2:
-1 = -1*(1, 1, 0) + 0*(0, 1, 1) + 0*(1, 0, 1)
2 = 0*(1, 1, 0) + 2*(0, 1, 1) + 0*(1, 0, 1)
So, (-1, 2, 0) can be written as a linear combination of the vectors in Set 2.
3 = 3*(1, 1, 0) + 0*(0, 1, 1) + 0*(1, 0, 1)
4 = 0*(1, 1, 0) + 4*(0, 1, 1) + 0*(1, 0, 1)
So, (3, 4, 0) can also be written as a linear combination of the vectors in Set 2.
Therefore, both sets can be used to span the set in R^3, and the vectors (-1, 2, 0) and (3, 4, 0) can be expressed as linear combinations of the vectors in each of these sets.
To find two sets of vectors that could be used to span a set in R^3, we need to find linearly independent vectors. Two such sets are the standard basis vectors and the ones given in the question.
1. Standard basis vectors:
The standard basis vectors in R^3 are:
i = (1, 0, 0)
j = (0, 1, 0)
k = (0, 0, 1)
These vectors form a basis for R^3 because any vector in R^3 can be written as a linear combination of these three vectors.
Now, let's show how the vectors (-1, 2, 0) and (3, 4, 0) can be written as linear combinations of the standard basis vectors:
(-1, 2, 0) = -1 * i + 2 * j + 0 * k = -i + 2j + 0k
(3, 4, 0) = 3 * i + 4 * j + 0 * k = 3i + 4j + 0k
2. Given vectors:
The vectors (-1, 2, 0) and (3, 4, 0) can also be used to span a set in R^3. To check if these vectors are linearly independent, we put them in a matrix and reduce it to row-echelon form.
Matrix equation:
[-1, 3]
[2, 4]
[0, 0]
Reducing to row-echelon form:
[1, -3]
[0, 10]
[0, 0]
Since there are no rows with all zeros, and every row has a leading 1, the given vectors are linearly independent.
Now, let's write the vectors (-1, 2, 0) and (3, 4, 0) as linear combinations of these vectors:
(-1, 2, 0) = -1 * (-1, 2, 0) + 0 * (3, 4, 0)
(3, 4, 0) = 0 * (-1, 2, 0) + 1 * (3, 4, 0)
In both cases, we are using one vector with a coefficient of 1 and the other vector with a coefficient of 0, as the vectors are linearly independent.