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Let $\mathbf{v}_{1}=\left[\begin{array}{r}{4} \\ {-3} \\ {7}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{1} \\ {9} \\ {-2}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}{7} \\ {11} \\ {6}\end{array}\right],$ and $H=$ Jse this information to find a basis for $H .$ There is more than one answer.

$\left\{v_{1}, v_{2}\right\}$ is one basis for $H$

Calculus 3

Chapter 4

Vector Spaces

Section 3

Linearly Independent Sets; Bases

Vectors

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for this example, you have been provided with three vectors that came from our three. And what we're going to do here is determined a vector space age by letting it be a span Northeast ban of the one V two and V three. So our goal here is to find a basis for this vectors, subspace h of our three, and what we would typically do is form a matrix, which would correspond to the columns of these three vectors. But let's suppose that were given some different information. Say, for example, that we know that four times V one plus five times V two minus three times V three is equal to zero where zero here is the zero vector. An equation like this allows us to write the following dependence relation. Let's add three times V three double sides than divide by three, and we see that the three is equal to 4/3 V one plus 5/3 V two. So the vector V three can be solved for in terms of the prior two vectors. Whatever this happens, we know that the set to be one V two V three is linearly dependent in this situation. So we want to basis, which means we need a linearly independent set so that Spans and Selena independent will know that's of a space ISS. So what we have to do here is take the vector v three and if we eliminate it, it will still spend a space age. We have that h is equal to the span of V one and V two, since a dependence relation for V three allows us to delete it from the set. Now we still have to determine if these vectors are linearly independent. That way we'll know if we have a basis or not. But when we consider the one of the two, let's look at the ones first entry one. If we multiply this entry by four, we'll get this corresponding entry. Multiply the next century by four. We do not get the century. This is enough to tell us. At the vectors. V one and V two are not multiples off each other, and since we have two vectors, not multiples of each other, this implies that be one. The two as a set is linearly independent, so let's pause for a moment. We have a set that spans. We deleted V three so that the new set, which is smaller, is now linearly independent. That allows us to write this conclusion. Therefore, the set the one be to forms a basis four and we have to be careful here. It does not form a basis for our three. It forms the basis for the said it spans so we can say it forms a basis for H and that completes this example.

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