404 CHAPTER 18. LINEAR FUNCTIONS
(c)
0 1 0 2 1 2 20 3 2 12 1 6 80 1 1 5 0 2 30 2 1 7 0 3 4
(d)
0 1 0 2 0 1 00 3 2 6 0 5 40 1 1 2 0 2 20 2 1 4 0 3 2
(e)
0 1 0 2 1 1 20 3 2 6 1 5 10 1 1 2 0 2 10 2 1 4 0 3 1
6. Suppose A is an m× n matrix. Explain why the rank of A is always no larger thanmin(m,n) .
7. A matrix A is called a projection if A2 = A. Here is a matrix. 2 0 21 1 2−1 0 −1
Show that this is a projection. Show that a vector in the column space of a projectionmatrix is left unchanged by multiplication by A.
8. Let H denote span((
12
),
(24
),
(13
)). Find the dimension of H and deter-
mine a basis.
9. Let H denote span
120
,
240
,
131
,
011
. Find the dimension of
H and determine a basis.
10. Let H denote span
120
,
140
,
131
,
011
. Find the dimension of
H and determine a basis.
11. Let M ={u= (u1,u2,u3,u4) ∈ R4 : u3 = u1 = 0
}. Is M a subspace? Explain.
12. Let M ={u= (u1,u2,u3,u4) ∈ R4 : u3 ≥ u1
}. Is M a subspace? Explain.
13. Let w ∈ R4 and let M ={u= (u1,u2,u3,u4) ∈ R4 : w ·u= 0
}. Is M a subspace?
Explain.
14. Let M ={u= (u1,u2,u3,u4) ∈ R4 : ui ≥ 0 for each i = 1,2,3,4
}. Is M a subspace?
Explain.
15. Let w,w1 be given vectors in R4 and define
M ={u= (u1,u2,u3,u4) ∈ R4 : w ·u= 0 and w1 ·u= 0
}.
Is M a subspace? Explain.
16. Let M ={u= (u1,u2,u3,u4) ∈ R4 : |u1| ≤ 4
}. Is M a subspace? Explain.