Span Of 3 Vectors In R3, The span of three vectors in R3 refers to the set of all possible linear combinations of these three, General, span-of-3-vectors-in-r3, Timnesia
The span of three vectors in R3 refers to the set of all possible linear combinations of these three vectors. In other words, if we have three vectors in R3, we can obtain all other vectors in R3 by scaling and adding these three vectors in different ways.
To understand this concept better, let's consider an example. Suppose we have three vectors in R3, namely,
v1 = (1, 2, 3)
v2 = (4, 5, 6)
v3 = (7, 8, 9)
To find the span of these three vectors, we need to find all possible linear combinations of these vectors. A linear combination of these vectors can be written as
c1v1 + c2v2 + c3v3
where c1, c2, and c3 are scalars. In other words, we can scale each vector by a scalar and add them together to obtain a new vector.
Let's consider some examples of linear combinations of these vectors.
- If we take c1 = 1, c2 = 0, and c3 = 0, we get the vector (1, 2, 3). This is just v1 itself.
- If we take c1 = 0, c2 = 1, and c3 = 0, we get the vector (4, 5, 6). This is just v2 itself.
- If we take c1 = 0, c2 = 0, and c3 = 1, we get the vector (7, 8, 9). This is just v3 itself.
These three vectors are called the "basis" vectors of the span of the three vectors. This is because any other vector in the span can be written as a linear combination of these three basis vectors.
To see this, let's consider another vector in R3, say,
u = (2, 3, 4)
We want to find out if this vector is in the span of v1, v2, and v3. To do this, we need to find scalars c1, c2, and c3 such that
c1v1 + c2v2 + c3v3 = u
We can write this as a system of linear equations:
c1 + 4c2 + 7c3 = 2
2c1 + 5c2 + 8c3 = 3
3c1 + 6c2 + 9c3 = 4
We can solve this system of equations to get
c1 = -1
c2 = 2
c3 = -1
Therefore,
-1v1 + 2v2 - 1v3 = (2, 3, 4)
This means that u is a linear combination of v1, v2, and v3.
In general, if we have n vectors in R3, the span of these vectors is a subspace of R3 that contains all possible linear combinations of these vectors. The dimension of the span is the number of linearly independent vectors in the set. If all n vectors are linearly independent, the span is a three-dimensional subspace of R3. If some vectors are linearly dependent, the span is a lower-dimensional subspace of R3.
In conclusion, the span of three vectors in R3 is the set of all possible linear combinations of these vectors. The basis vectors of the span are the three original vectors themselves, and any other vector in the span can be written as a linear combination of these basis vectors. The span is a three-dimensional subspace of R3 if the three vectors are linearly independent.