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[answered] Math 110 Homework Assignment 9 due date: Nov. 18, 2016 1. (

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:

A

C=

,

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

n

o

Ker(A) = ~v ? Rn | A~v = ~0 .

While n

o

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).

4.

(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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