[answered] Math 110 Homework Assignment 11 due date: 2, 2016 1. The li

Question 1 and 2?

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Math 110 Homework Assignment 11

 

due date: Dec. 2, 2016 1. The linear transformation T 2 0

 

6 A=

 

3 1 11

 

?3 0 ?9 : R4 ?? R3 is given by the matrix 1

 

1 0 3 0

 

0 , which has RREF 0 1 2 0 .

 

1

 

0 0 0 1 (You don?t have to prove that this is the RREF.)

 

(a) Find a basis for the image of T . 5

 

(b) The vector w

 

~ = 5 is in the image. Find the linear combination of basis vectors

 

0

 

from (a) which gives w.

 

~ Use this to find a vector ~v in R4 with T (~v) = w.

 

~

 

(c) Find a basis for the kernel of T .

 

(d) Write down all the solutions to T (~v) = w,

 

~ with w

 

~ the vector from part (b). You

 

shouldn?t have to do any complicated calculations to figure this out.

 

(e) Find all solutions to the system of equations

 

2x +

 

+

 

6z + w = 5

 

3x + y + 11z

 

= 5

 

?3x

 

+ ?9z + w = 0

 

using the ?old? method ? the RREF method of solving equations that we learned

 

in the first weeks of class.

 

(f) Explain the connection between (d) and (e). 2. Let V ? R4 be the subspace spanned by ~v1 = (2, 1, 3, 1) and ~v2 = (5, 3, 2, 1), and set

 

w

 

~ 1 = (1, 2, 5, 1), w

 

~ 2 = (8, 7, ?1, 0), w

 

~ 3 = (3, 1, 2, 2).

 

(a) Is w

 

~2 ? w

 

~ 1 a linear combination of ~v1 and ~v2 ?

 

(b) Does w

 

~1 + V = w

 

~2 + V ?

 

(c) Does w

 

~1 + V = w

 

~3 + V ? 1 3. Suppose that T1 : Rn ?? Rm and T2 : Rm ?? Rp are linear transformations. In this

 

question we will practice thinking about compositions by thinking about the kernel of

 

T2 ? T1 .

 

(a) Prove that ~v ? Ker(T2 ? T1 ) if and only if T1 (~v ) ? Ker(T2 ).

 

(b) Show that Ker(T1 ) ? Ker(T2 ? T1 ).

 

(c) If T2 is injective, show that Ker(T1 ) = Ker(T2 ? T1 ).

 

(d) Prove that T2 ? T1 = 0 if and only if Im(T1 ) ? Ker(T2 ). (Here 0 means the zero

 

linear transformation ? the map that sends each ~v ? Rn to ~0 ? Rp .)

 

(e) Suppose that T : R6 ?? R6 is a linear transformation, and that T ? T = 0. Prove

 

that dim(Im(T )) 6 3. (Suggestion: Combine (d), the Rank-Nullity theorem,

 

and H10 Q1(e).)

 

Note: Here are some basic facts about making arguments about sets. Suppose that X

 

and Y are sets.

 

? If you are trying to prove that X ? Y then this is the same as showing that for

 

each x ? X, it?s also true that x ? Y . I.e., if, no matter which element x ? X

 

you?re given, you can show that this element is also in Y , then you?ve shown that

 

X ?Y.

 

? If you are trying to prove that two sets X and Y are equal, one way is to first prove

 

that X ? Y and then prove that Y ? X. If both of these are true, then X = Y .

 

You can try the method above for proving each direction of the containments.

 

4. In this problem we will continue thinking about compositions, kernels, and invertibility.

 

(a) If A is an m ? n matrix, explain why Rank(A) 6 n and Rank(A) 6 m (and so

 

Rank(A) 6 min(m, n)).

 

(b) Let T : R6 ?? R4 be a linear transformation. Prove that dim(Ker(T )) > 2.

 

Suppose that A is a 3 ? 2 matrix, and B is a 2 ? 3 matrix.

 

(c) Explain why the 3 ? 3 matrix AB can never be invertible. (Hint: what can you

 

say about Ker(AB)?)

 

(d) Find an example of matrices A and B where the 2 ? 2 matrix BA is invertible.

 

(e) If BA is invertible, what must be the dimension of Ker(A), and what must be the

 

dimension of Ker(B)? Explain why.

 

2

 

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