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
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 .
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
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
6z + w = 5
3x + y + 11z
+ ?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
~ 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
? 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.
This question was answered on: Sep 18, 2020
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