# MATB24H3 Lecture Notes - Lecture 5: Linear Map, Scalar Multiplication, Binary Operation

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Published on 7 Oct 2020

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Week 1: (real) Vector Space & binary operation

Example Questions:

1. State the definition of a real vector space. [basic definition]

2. Prove the zero element in a real vector space is unique. [proof by definition]

3. FB page 189, question #3

[example of real vector space with non-standard addition and scalar multiplication]

4. FB page 189, question #17 [find non-standard zero element use the definition]

5. FB page 190, question #23/25 [proof using the uniqueness of zero element]

6. State the definition of a binary operator and the definition of a commutative or associative.

[definition of binary operation]

7. Give an example of a binary operator in {1, 2, 3} such that the operator is commutative? Such

that the operator is associative? [definition of binary operation]

8. SA page 17, prove (-1)v = -v for all v [proof by definition]

9. SA page 17, question #6 [definition of vector space, but a bit complex]

Week 2: Concept of Vector Space & Subspace

Example Questions:

1. State the definition of a subspace S in vector space V.

2. State the definition of vectors {v1, v2, …, vn} are linearly independent to each other. State

the definition of a linear combination of {v1, v2, …, vn}. State the definition of sp{v1, v2, …, vn}

3. Give an example of a subspace in R^3 other than R^3 itself. Give an example of a subspace

in P^2 other than P^2 itself. [definition]

4. FB page 202, question #5, 6 [Prove by definition of subspace]

5. FB page 202, question #8, 9 [Prove by definition of span] [Follow the hint, how to prove two

subspace are equal]

6. FB page 202, question #11, 16 [Prove linear independence]

7. FB page 203, question #27 [subspace proving]

8. SA page 25, question 9 [subspace proving about functions]

Week 3: Concept of Vector Space cont.

Example Questions:

1. State the definition of a set B = {v1, v2,..., vn} is a basis of vector space V.

2. Proof if B = {v1, v2, …, vn} is a basis of V, then any vector w in V can be expressed as a

unique linear combination of B. [uniqueness of basis’s linear combination]

3. Proof if {v1, v2, v3} is a linear dependent set in V, then sp{v1, v2, v3} = sp{v1, v2}. Further,

proof if {v1, v2, …, vn} is a linearly dependent set, then sp{v1, v2, …, vn-1} = sp{v1, v2, …,

vn-1, vn} [reduction lemma]

4. FB page 203, question #29 [basis proving]

5. FB page 203, question #34 [basis proving with subspace calculation, a bit complex]

6. FB page 203, question #32 [basis proving using definition]

7. SA page 37, question#10, 11 [linear independence and spanning proving by defintion]

8. SA page 38, question#12, 13 [example of dim(LI set) <= dim(V) <= dim(spanning set)]