7 - bases and dimension Exam 2026
Questions and Answers Graded A+
what is a finite-dimensional vector space? - Correct answer-a vector space is said
to be finite-dimensional if it has a finite basis
let V be a vector space and let S⊆V be a finite spanning set. what do we know if
L⊆V? - Correct answer-- if L⊆V is linearly independent then L is finite and
|L| ≤ |S|
how do you prove this? - Correct answer-proof in notes
what is the basis theorem? - Correct answer-- let V be a fin-dim vector space
- if B and C are both bases of V then B and C are finite sets and |B| = |C|
how do you prove the basis theorem? - Correct answer-- since V is fin-dim, it has a
finite basis (in particular a finite spanning set)
- so all bases of V are finite
- since B is a linearly independent set and C is a spanning set, we know that |B| ≤
|C|
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, - since B is a spanning set and C is a linearly independent set, we know that |C| ≤
|B|
- therefore |B| = |C|
what is the dimension of a vector space? - Correct answer-- let V be a fin-dim
vector space with basis B
- the dimension of V is dimV = |B|
what does it mean for a vector space to be infinite-dimensional? - Correct answer--
a vector space which has no finite spanning set is said to be infinite-dimensional
what does it mean for a vector space to contain an infinite linearly independent set
? - Correct answer-- if a vector space V contains an infinite linearly independent
set L, then V is infinite-dimensional
what is an example of an infinite-dimensional vector space? - Correct answer-the
vector space V=P of polynomials is infinite-dimensional because it contains the
infinite linearly independent subset:
L = {1,x^2,x^3,x^4,...}
give an example of working out the dimension of a vector space - Correct answer--
let n be a positive integer and let K be a field
- the vector space K^n has basis {e1,e2,...,en}
©COPYRIGHT 2025,ALL RIGHTS RESERVED 2
Questions and Answers Graded A+
what is a finite-dimensional vector space? - Correct answer-a vector space is said
to be finite-dimensional if it has a finite basis
let V be a vector space and let S⊆V be a finite spanning set. what do we know if
L⊆V? - Correct answer-- if L⊆V is linearly independent then L is finite and
|L| ≤ |S|
how do you prove this? - Correct answer-proof in notes
what is the basis theorem? - Correct answer-- let V be a fin-dim vector space
- if B and C are both bases of V then B and C are finite sets and |B| = |C|
how do you prove the basis theorem? - Correct answer-- since V is fin-dim, it has a
finite basis (in particular a finite spanning set)
- so all bases of V are finite
- since B is a linearly independent set and C is a spanning set, we know that |B| ≤
|C|
©COPYRIGHT 2025,ALL RIGHTS RESERVED 1
, - since B is a spanning set and C is a linearly independent set, we know that |C| ≤
|B|
- therefore |B| = |C|
what is the dimension of a vector space? - Correct answer-- let V be a fin-dim
vector space with basis B
- the dimension of V is dimV = |B|
what does it mean for a vector space to be infinite-dimensional? - Correct answer--
a vector space which has no finite spanning set is said to be infinite-dimensional
what does it mean for a vector space to contain an infinite linearly independent set
? - Correct answer-- if a vector space V contains an infinite linearly independent
set L, then V is infinite-dimensional
what is an example of an infinite-dimensional vector space? - Correct answer-the
vector space V=P of polynomials is infinite-dimensional because it contains the
infinite linearly independent subset:
L = {1,x^2,x^3,x^4,...}
give an example of working out the dimension of a vector space - Correct answer--
let n be a positive integer and let K be a field
- the vector space K^n has basis {e1,e2,...,en}
©COPYRIGHT 2025,ALL RIGHTS RESERVED 2
