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Let $W$ be the set of all vectors of the form shown, where $a, b,$ and $c$ represent arbitrary real numbers. In each case, either find a set $S$ of vectors that spans $W$ or give an example to show that $W$ is not a vector space.$\left[\begin{array}{c}{a-b} \\ {b-c} \\ {c-a} \\ {b}\end{array}\right]$

$W$ is a span of set $\left\{\left[\begin{array}{c}{1} \\ {0} \\ {-1} \\ {0}\end{array}\right],\left[\begin{array}{c}{-1} \\ {1} \\ {0} \\ {1}\end{array}\right],\left[\begin{array}{c}{0} \\ {-1} \\ {1} \\ {0}\end{array}\right]\right\}$

Calculus 3

Chapter 4

Vector Spaces

Section 1

Vector Spaces and Subspaces

Vectors

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Hello. For this question, we need to either show that w can be written as this span of a set of vectors or show that it can. It cannot be so. Our eventual goal is toe have W as the span of some set of vectors. So when they use V, you and W But we know that if this were to be true, there would have to be three vectors here because we have three arbitrary constants. A be and say now, Ah, remembering what the definition of a span is is just all vectors of this form, and I'm gonna, uh, use the constants given in the problem. So if w can be written as this span of some set of vectors, they must be able to break this form into the sum of three vectors multiplied by the constants A b and safe. So I eventually want this, and now I start eyeballing If I look at this, uh, this vector in the first lot, I have one times in a so that means I have to put a one here. I have no A's in the second or fourth slot, but I have a negative a in the third slot. Now, here on the second vector, I have a negative B in the first slot, positive being the second slot. None in the next two. On following the same procedure, I have zero in the first lot, a negative, see in the second and then sees in the 3rd and 4th slots. So you've successfully broken up this form into an arbitrary, constant times a vector which Aiken now label Well, V was was my first one. A na arbitrary, constant times a second vector, which I can now relabel w then arbitrary, constant times. The third vector, which I can Now, uh, this is you. And then I can label this as w So we have shown that this, uh, this set is actually the span of a set of vectors v u and W, which are 10 negative. 10 negative. 1100 and zero. Negative. 111 And we're done

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