[Solved] MATH141: Linear Analysis I Homework 08

Construct examples of linear transformation that satisfy the following requirements. If no such examples are possible, explain why. (Hint: Problems #6-8 of Homework 06 help you connect one-to-one or onto linear transformations to properties of matrices.)

2. (Strang 2.1 #2) Which of the following subsets of R³ are actually subspaces? For each subspace you find, find a basis for that subspace. Describe your reasoning.
   • The plane of vectors with first component b₁ = 0.
   • The plane of vectors^~b with first component b₁ = 1.
   • The vectors ^~b with b₂b₃ = 0 (notice that this is the union of two subspaces, the plane b₂ = 0 and the plane b₃ = 0).
   • All linear combinations of two given vectors and .
   • The plane of vectors^~b that satisfy b₃ b₂ + 3b₁ = 0.
3. Determine each of the following statements true or false. Explain your reasoning.
   • {^~0} is a vector subspace of any Rⁿ, where^~0 has n zeroes as coordinates.
   • Any straight line in R² is a vector subspace of R².
   • Any two-dimensional plane going through the origin in R³ is a vector subspace of R³.
4. (added on Wednesday) Finish the worksheet in lecture titled Basis for N(A) and C(A), a copy of which is posted on CatCourses. Turn in a digital copy of your solution together with the rest of this homework set, and bring a hard copy of your solutions to class on Monday.
5. (revised on Wednesday) The dimension of a vector subspace W, denote by dimW, is defined to be the number of vectors in its basis.
   • For the matrix in the worksheet,. what is dimN(A)? What is dimC(A)?
   • If A is an m-by-n matrix with rank r. What is dimN(A)? What is dimC(A). Explain your reasoning. (Hint: review the worksheet.)
6. (postponed to next week) T : Rⁿ R^m is a linear transformation.
   • Is ker(T) a subspace of Rⁿ?. Explain your reasoning. If yes, how can you find a basis for ker(T)?
   • Is range(T) a subspace of R^m?. Explain your reasoning. If yes, how can you find a basis for range(T)?

(Hint: Connect ker(T) and range(T) to column space and nullspace of some matrix.)

MATH 141: Linear Analysis I Homework 08 Fall 2019

7. Follow the steps below to prove the theorem: If {~e₁,~e₂,,~e_n} is a basis for Rⁿ, then any vector ~x in Rⁿ can be written as a linear combination of ~e₁,~e₂,,~e_n in a unique way.
   • Which requirement for {~e₁,~e₂,,~e_n} to be a basis ensures that ~x can be written as some linear combination of ~e₁,~e₂,,~e_n?
   • Suppose that ~x can be written as a linear combination of ~e₁,~e₂,,~e_n in two different ways. That is,

~x = c₁~e₁ + c₂~e₂ + + c_n~e_n, and ~x = d₁~e₁ + d₂~e₂ + + d_n~e_n

where all the cs are not the same as all the ds. By calculating ~x ~x, show that one requirement for {~e₁,~e₂,,~e_n} to be a basis has been violated.

• Explain briefly why putting parts (a) and (b) together leads to a proof of the theorem.