## 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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