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Let $H$ be the set of all vectors of the form $\left[\begin{array}{c}{s} \\ {3 s} \\ {2 s}\end{array}\right] .$ Find a vector $\mathbf{v}$ in $\mathbb{R}^{3}$ such that $H=\operatorname{Span}\{\mathbf{v}\} .$ Why does this show that $H$ is a subspace of $\mathbb{R}^{3} ?$

Thus, H is a subspace of $R^{3}$

Calculus 3

Chapter 4

Vector Spaces

Section 1

Vector Spaces and Subspaces

Vectors

Campbell University

Harvey Mudd College

Baylor University

Boston College

Lectures

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In mathematics, a vector (…

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Hello. For this problem, we need to show that we can write this vector as a if I call the set H, which is just the set of all vectors of this form or s is some arbitrary element of our We need to show that we can write H as this span of some vector the which will also show that h is a subspace of our three. The reason why this shows that h is a subspace of our three comes from the definition off span. If you if you remember the conditions that a set has to meet to be a subspace if if If all of the elements are members of a larger vector space, those air exactly the conditions is there exactly what defines a span of the of a given vector. So this one is relatively simple because remember, the span of a single vector is just all arbitrary constants s in our field. In this case, the real numbers times that vector. If we look here, we can immediately just factor out in s and get that h is equal toe s times. The vector one, 32 again s is just some arbitrary, constant, so h is equal to the span of our vector. 132 The notation gets a little bit sloppy when you're actually doing with real vectors. But this just shows that this is just a non arbitrary, constant times this given vector on we're done.

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