Math 110 Homework Assignment 9
due date: Nov. 18, 2016
Math 110 Homework Assignment 9
due date: Nov. 18, 2016 1.
(a) Prove or disprove: the vectors ~v1 = (2, 3) and ~v2 = (1, 5) form a basis for R2 .
(b) Prove or disprove: the vectors w
~ 1 = (1, 9) and w
~ 2 = (2, 3) form a basis for R2 .
(c) How many bases can a subspace have?
(d) Let V be the set of vectors (x, y, z, w) in R4 which are the solutions to the equations:
x + 0y + 3z ? 2w = 0
0x + y ? 4z ? 9w = 0
The subset V is a subspace of R4 . Find a basis for V .
(Suggestion: V is given as the set of solutions to a system of linear equations.
You know how to parameterize all the solutions. . . ) 2. Let ~v1 ,. . . , ~vk be vectors in Rn , and let A = [~v1 | ~v2 | ? ? ? ~vk ], i.e., the matrix
whose columns are ~v1 ,. . . , ~vk . Prove that ~v1 ,. . . , ~vk are linearly independent if and only
if Rank(A) = k.
Reminder: Proving a statement with an ?if and only if? requires proving both directions. You assume that Rank(A) = k and then deduce that ~v1 ,. . . , ~vk are linearly
independent. Then assume that ~v1 , . . . , ~vk are linearly independent and prove that
Rank(A) = k. (If you can do both steps at the same time that is fine too.) One other
reminder: looking for c1 ,. . . , ck so that c1~v1 + ? ? ? + ck~vk = ~0 is the same as solving a
system of linear equations.
3. Linear transformation puzzlers
(a) Consider a linear transformation T : Rn ?? Rm . If ~v1 ,. . . , ~vk are linearly dependent vectors in Rn , are the vectors T (~v1 ),. . . , T (~vk ) necessarily linearly dependent
in Rm ? If so, why?
(b) If A is an n ? p matrix, and B is a p ? m matrix, with Im(B) ? Ker(A), what can
you say about the product AB?
1 (c) if A is a p ? m matrix, and B a q ? m matrix, and we make a (p + q) ? m matrix
C by ?stacking? A on top of B:
what is the relation between Ker(A), Ker(B), and Ker(C)? Note: So far, we have only used the symbols Ker and Im when talking about a linear
transformation T . In this homework we?re going to extend this notation and also use
Ker and Im when talking about a matrix. If A is an m ? n matrix, then
Ker(A) = ~v ? Rn | A~v = ~0 .
Im(A) = w
~ ? Rm | there is a v ? Rn so that A~v = w
~ . The connection between this notation and our usual notation about linear transformations is that if T : Rn ?? Rm is a linear transformation and A the standard matrix of
T , then Ker(T ) = Ker(A) and Im(T ) = Im(A).
(a) Suppose that we have a system of linear equations in n variables. For instance,
we might have m equations: a11 x1 + a12 x2 + ? ? ? + a1n xn = 0
a21 x1 + a22 x2 + ? ? ? + a2n xn = 0
am1 x1 + ak2 x2 + ? ? ? + amn xn = 0
where the aij are any numbers in R. Show that the set of solutions to this system
of equations forms a subspace of Rn .
(b) The vectors ~v1 = (?1, 3, 1, 2), ~v2 = (2, 3, 2, ?7), and ~v3 = (2, 1, 1, ?6) span a 3dimensional subspace of R4 . Find a single equation of the form ax+by+cz+dw = 0
whose solutions are this subspace. 2
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