MAT 240 MAT 240 MAT 240 - A SSIGNMENT SSIGNMENT SSIGNMENT # 4 D UE MARCH 21

Instructions

Please submit solutions to the questions listed below on Crowdmark. If you did not receive an email

from CM with this assignment, contact J. Thind. Your assignment is due before 11.59pm on Sunday,

March 21.

Full justification is required for all questions. Don’t forget - proofs have words, not just strings of

calculations/algebraic manipulations.

Unless otherwise indicated, F denotes a field and V denotes a vector space over F.

Assigned Questions

(1) Let V = Sym2(F) (and assume that char(F) 6= 2).

(a) Find Q - the change of coordinate matrix from β to γ coordinates.

(b) Find Q?1

- the change of coordinate matrix from γ to β coordinates.

(c) Find [T]β and [T]γ.

(2) Let A ∈ Mn×n(F) be a matrix with columns c1, . . . , cn.

(a) Prove that the columns of A form a basis for F

n

if and only if A is invertible.

(b) Deduce that for every invertible matrix Q there exist bases β, γ of F

n

so that Q is the

change of coordinate matrix from β to γ coordinates.

(c) (T/F:) Suppose that V is any n-dimensional vector space and T : V → V an isomorphism.

If Q is invertible, then there exist bases β, γ of V so that [T]

γ

β = Q.

(d) (T/F:) Suppose that V is any n-dimensional vector space and T : V → V an isomorphism.

If Q is invertible, then there exists a basis β of V so that [T]β = Q.Find the R - the RREF of A, and express R = QA for some invertible

matrix Q.

(Hint: Keep track of your row operations, and recall that every invertible matrix is a product

of elementary matrices.)

MAT 240 MAT 240 MAT 240 - A SSIGNMENT SSIGNMENT SSIGNMENT # 4 D UE MARCH 21

(4) Prove that for every n × n matrix A there exists an invertible matrix Q so that QA is uppertriangular.

(Hint: Use induction on n and row-operations/elementary matrices.)

(5) Define an equivalence relation on Mm×n(F) by A ～ B if there exists invertible matrices

Q ∈ Mm×m(F), P ∈ Mn×n(F) so that A = QBP.

(a) Prove that this is an equivalence relation.

(b) Express the number of distinct equivalence classes in Mm×n(F) as a function of m and n.

(And, as usual, prove your claim.)

(6) (a) Let T : V → W, S : W → X be isomorphisms. Prove that S ? T is an isomorphism and

find a formula for (S ? T)

?1

.

(b) Prove that the product of two invertible matrices of the same size is invertible using (a).

(c) Give a second proof of the fact that the product of two invertible matrices of the same

size is invertible, using elementary matrices.

(d) Give a third proof of the fact that the product of two invertible matrices of the same size

is invertible, using rank.

(7) Find an isomorphism T : Sk3(R) → W, where W = {(x, y, z, w) ∈ R

4

| x + y + z + w = 0}.

(As always, justify your claim!)

(8) Let T : R

2 → R

2 be a linear transformation so that T(1, 1) = (2, 2) and T(1, 0) = (?1, 0). Let

γ = {(1, 1),(1, 0)} and β be the standard basis.

(a) Find [T]γ.

(b) Find Q - the change of coordinate matrix from γ to β coordinates.

(c) Find Q?1 using row operations on (Q|I2).

(d) Find [T]β and deduce an explicit formula for T(x, y).

(e) Which coordinates are best suited for studying this transformation - β or γ? Why?

(9) Determine if the statements below are true or false. If true, give a proof. If false, explain why,

and/or provide a counterexample.

(a) There exists a m × n matrix A so that the system Ax = b has no solutions for all b ∈ F

m.

(b) Suppose that A ∈ Mn×n(F). If Ax = b has solutions for all b ∈ F

n

, then A is invertible.

(c) If A ∈ Mm×n(F) has rank n ? 1, then Ax = b has no solutions for some b ∈ F

m.

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