Let
v1 = (1, 0, −1)
v2 = (0, 2, 2)
v3 = (−3, 4, 7)
and let W = Span{v1, v2, v3}.
Show that W = Span{v1, v2}?
How would I go by doing this?
Thanks in advance.
isnt 2v2-3v1=v3 ?
Yes
To show that W = Span{v1, v2}, we need to show that every vector in W can be expressed as a linear combination of v1 and v2, and every vector in Span{v1, v2} is also in W.
Let's start by finding the set of vectors that can be expressed as a linear combination of v1 and v2. We can do this by finding the span of v1 and v2.
The span of v1 and v2 is a set of all possible linear combinations of v1 and v2. In other words, it is the set of all vectors that can be written as:
c1*v1 + c2*v2
where c1 and c2 are scalars.
So, let's find the span of v1 and v2:
span{v1, v2} = {(c1*v1 + c2*v2) | c1, c2 ∈ ℝ}
To show that W = Span{v1, v2}, we need to prove that every vector in W is in span{v1, v2}, and every vector in span{v1, v2} is in W.
1. Every vector in W is in span{v1, v2}:
This can be shown by expressing each vector in W as a linear combination of v1 and v2. If every vector in W can be expressed in this way, then it is in span{v1, v2}.
2. Every vector in span{v1, v2} is in W:
This can be shown by expressing each vector in span{v1, v2} as a linear combination of v1, v2, and v3. If every vector in span{v1, v2} can be expressed in this way, then it is in W.
So, by proving both of these statements, we can conclude that W = span{v1, v2}.
To explicitly show that every vector in W can be expressed as a linear combination of v1 and v2, we need to solve the equation:
c1*v1 + c2*v2 + c3*v3 = w
where w is an arbitrary vector in W.
In this case, we have:
w = c1*v1 + c2*v2 + c3*v3
Now, substitute the values of v1, v2, and v3:
w = c1*(1, 0, -1) + c2*(0, 2, 2) + c3*(-3, 4, 7)
Simplifying, we get:
w = (c1, 0, -c1) + (0, 2c2, 2c2) + (-3c3, 4c3, 7c3)
Combining like terms, we have:
w = (c1-3c3, 2c2+4c3, -c1+2c2+7c3)
Now, we need to find the values of c1, c2, and c3 such that w satisfies the equation. This can be done by solving the system of equations:
c1 - 3c3 = w1
2c2 + 4c3 = w2
-c1 + 2c2 + 7c3 = w3
where w1, w2, and w3 are the components of the vector w.
By solving this system of equations, we can find the values of c1, c2, and c3. If there exist values that satisfy the equations, then every vector w in W can be expressed as a linear combination of v1 and v2, which means W ⊆ span{v1, v2}.
Finally, we also need to show that span{v1, v2} ⊆ W. We can do this by showing that every vector in span{v1, v2} can be expressed as a linear combination of v1, v2, and v3. If this is true, then every vector in span{v1, v2} is also in W.
To summarize, to show that W = Span{v1, v2}, you need to prove:
1. Every vector in W is in span{v1, v2}.
2. Every vector in span{v1, v2} is in W.
To do this, you need to find the set of vectors that can be expressed as linear combinations of v1 and v2, and then show that every vector in W can be expressed as a linear combination of v1 and v2, and vice versa. You can do this by solving the necessary equations as explained above.