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MATH141-Homework 8 Solved
1. 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.)
one-to-one but not onto
onto but not one-to-one
both one-to-one and onto
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.
(a) The plane of vectors with first component b1 = 0.
(b) The plane of vectors~b with first component b1 = 1.
(c) The vectors ~b with b2b3 = 0 (notice that this is the union of two subspaces, the plane b2 = 0 and the plane b3 = 0).
(d) All linear combinations of two given vectors and .
(e) The plane of vectors~b that satisfy b3 − b2 + 3b1 = 0.
3. Determine each of the following statements true or false. Explain your reasoning.
(a) {~0} is a vector subspace of any Rn, where~0 has n zeroes as coordinates.
(b) Any straight line in R2 is a vector subspace of R2.
(c) 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.
(a) For the matrix in the worksheet, . what is dimN(A)? What is dimC(A)?
(b) 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 → Rm is a linear transformation.
(a) Is ker(T) a subspace of Rn?. Explain your reasoning. If yes, how can you find a basis for ker(T)?
(b) Is range(T) a subspace of Rm?. 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,...,~en} is a basis for Rn, then any vector ~x in Rn can be written as a linear combination of ~e1,~e2,...,~en in a unique way.
(a) Which requirement for {~e1,~e2,...,~en} to be a basis ensures that ~x can be written as some linear combination of ~e1,~e2,...,~en?
(b) Suppose that ~x can be written as a linear combination of ~e1,~e2,...,~en in two different ways. That is,
~x = c1~e1 + c2~e2 + ··· + cn~en, and ~x = d1~e1 + d2~e2 + ··· + dn~en
where all the c’s are not the same as all the d’s. By calculating ~x − ~x, show that one requirement for {~e1,~e2,...,~en} to be a basis has been violated.
(c) Explain briefly why putting parts (a) and (b) together leads to a proof of the theorem.
one-to-one but not onto
onto but not one-to-one
both one-to-one and onto
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.
(a) The plane of vectors with first component b1 = 0.
(b) The plane of vectors~b with first component b1 = 1.
(c) The vectors ~b with b2b3 = 0 (notice that this is the union of two subspaces, the plane b2 = 0 and the plane b3 = 0).
(d) All linear combinations of two given vectors and .
(e) The plane of vectors~b that satisfy b3 − b2 + 3b1 = 0.
3. Determine each of the following statements true or false. Explain your reasoning.
(a) {~0} is a vector subspace of any Rn, where~0 has n zeroes as coordinates.
(b) Any straight line in R2 is a vector subspace of R2.
(c) 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.
(a) For the matrix in the worksheet, . what is dimN(A)? What is dimC(A)?
(b) 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 → Rm is a linear transformation.
(a) Is ker(T) a subspace of Rn?. Explain your reasoning. If yes, how can you find a basis for ker(T)?
(b) Is range(T) a subspace of Rm?. 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,...,~en} is a basis for Rn, then any vector ~x in Rn can be written as a linear combination of ~e1,~e2,...,~en in a unique way.
(a) Which requirement for {~e1,~e2,...,~en} to be a basis ensures that ~x can be written as some linear combination of ~e1,~e2,...,~en?
(b) Suppose that ~x can be written as a linear combination of ~e1,~e2,...,~en in two different ways. That is,
~x = c1~e1 + c2~e2 + ··· + cn~en, and ~x = d1~e1 + d2~e2 + ··· + dn~en
where all the c’s are not the same as all the d’s. By calculating ~x − ~x, show that one requirement for {~e1,~e2,...,~en} to be a basis has been violated.
(c) Explain briefly why putting parts (a) and (b) together leads to a proof of the theorem.
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