## Linear Algebra and Its Applications, Exercise 2.3.6

Exercise 2.3.6. What are the geometric entities (e.g., line, plane, etc.) spanned by the following sets of vectors:

a) $(0, 0, 0)$, $(0, 1, 0)$, and $(0, 2, 0)$

b) $(0, 0, 1)$, $(0, 1, 1)$, and $(0, 2, 1)$

c) the combined set of six vectors above (which vectors are a basis for the space spanned?)

d) all vectors with positive components

Answer: a) Both the first vector and the third vector are a linear combination of the second vector: we have $(0, 0, 0) = 0 \cdot (0, 1, 0)$ and $(0, 2, 0) = 2 \cdot (0, 1, 0)$ so that the space spanned by the set of vectors is simply the line passing through the origin and the point $(0, 1, 0)$. This line is the $y$ axis in $\mathbf{R}^3$.

b) Since all of the vectors have the first ( $x$) component to be zero they are in the $yz$ plane. There is no nonzero $c$ for which $(0, 1, 1) = c (0, 0, 1)$ so those two vectors are linearly independent; however the third vector can be expressed as a nontrivial linear combination of the first two: $(0, 2, 1) = 2(0, 1, 1) - (0, 0, 1)$. Together the first two vectors span the $yz$ plane.

c) The vectors from (a) can all be expressed as linear combinations of the first two vectors from (b): $(0, 0, 0) = 0 (0, 1, 1)$, $(0, 1, 0) = (0, 1, 1) - (0, 0, 1)$, and $(0, 2, 0) = 2(0, 1, 1) - 2(0, 0, 1)$. Thus the space spanned by all six vectors is the same as that spanned by the vectors in (b), namely the $yz$ plane, and the first two vectors from (b) are a basis for the space.

d) Among the vectors with positive components are $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. These three vectors span $\mathbf{R}^3$ and serve as a basis for the space. So the set of all vectors with positive components spans $\mathbf{R}^3$.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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