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Chapter 1: Problem 11

Prove that \(\operatorname{span}(\\{x\\})=\\{a x: a \in F\\}\) for any vector \(x\) in a vector space, Interpret this result geometrically in \(\mathrm{R}^{3}\).

Short Answer

Expert verified
To prove that the span of a set containing a single vector x equals the set of all scalar multiples of that vector x, we showed two things: (1) Any ax, where a 鈭� F, is in the span of {x}, and (2) Any vector in the span of {x} can be written as ax, where a 鈭� F. We demonstrated that these conditions hold, implying that span({x}) = {ax: a 鈭� F}. Geometrically, in 鈩澛�, the span of a vector x represents a line passing through the origin and pointing in the direction of x, and all scalar multiples of x lie on this line.

Step by step solution

01

Show that ax 鈭� span({x}) for all a 鈭� F

To show that any ax is in the span of {x}, we can simply write ax as a linear combination of the vector x. Since the span is defined as the set of all linear combinations of the vectors in the set, it means that ax 鈭� span({x}) for all a 鈭� F by definition.
02

Show that any vector in span({x}) can be written as ax for some a 鈭� F

Now, let's take an arbitrary vector v in the span of {x}. By definition of span, we can write v as a linear combination of the vector x. In this case, since the set only contains the single vector x, the linear combination would simply be v = ax, where a 鈭� F. So any vector in the span of {x} can indeed be written as ax for some a 鈭� F.
03

Prove the equality of the sets

Since we have shown that: 1. Any ax, where a 鈭� F, is in the span of {x}, and 2. Any vector in the span of {x} can be written as ax, where a 鈭� F This implies that the set of all scalar multiples of vector x (that is, {ax: a 鈭� F}) is equal to the span of {x}. Therefore, we can conclude that span({x}) = {ax: a 鈭� F}.
04

Geometric interpretation in 鈩澛�

For the geometric interpretation in 鈩澛�, consider a particular vector x = [x鈧�, x鈧�, x鈧僝 in the space 鈩澛�. The span of this vector would be all the scalar multiples of x. Geometrically, this represents a line in 鈩澛� that passes through the origin and points in the direction of vector x. Each scalar multiple simply stretches or compresses the line but remains on the path defined by the original vector.

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Most popular questions from this chapter

Find the equation of the plane \(H:\) (a) with normal \(\mathbf{N}=3 \mathbf{i}-4 \mathbf{j}+5 \mathbf{k}\) and containing the point \(P(1,2,-3)\) (b) parallel to \(4 x+3 y-2 z=11\) and containing the point \(Q(2,-1,3)\)

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