# 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 basiss 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)]
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