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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.$\left[\begin{array}{c}{3 a+b} \\ {4} \\ {a-5 b}\end{array}\right]$

Thus, the set of vectors $W=\left[\begin{array}{c}{3 a+b} \\ {4} \\ {a-5 b}\end{array}\right]$ cannot form a vector space.

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 demonstrate that the set W which is the set of all vectors of this form, where nbr arbitrary real numbers is a vector space. Or we need to show that it's not, um, swell or it's going to be that it's not. And the the key issue here is the second slot, which is fixed at four, which means that we don't have a non arbitrary constant in the second slot. So all vectors in the space need to have four in the second slot. But if vector spaces are closed under multiplication by arbitrary constants. So, for example, if I have the vector zero for zero, which is within W, I just said and be equal to zero. Um, what we call this vector V just for convenience. And I multiply the by two when I get the vector 08 zero, which is not in W, which is which demonstrates the W is not a vector space because it is not closed under multiplication by arbitrary constants, and that's it

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