Download Linear Algebra Vocabulary Homework

Math 20610, Linear Algebra
Vocab Homework
Define the following terms:
1) Vector Space
2) Subspace
3) Linear Combination
4) Nontrivial Linear Combination
5) Span (noun)
6) Span (verb)
7) Linearly Independent
8) Matrix
9) Linear Transformation
10) Column Space
11) Null Space
12) Range
13) Kernel
14) Basis
15) Dimension
16) Rank
Every definition should include two separate parts:
a) An explanation of what kind of thing the term is. This explanation should make sure
to specifically mention any other things that are required in order for the term to have
meaning.
b) A definition of the term.
To get you started and to clarify the question, I have included answers to several of the questions.
Your homework should include answers to every question, including the ones where I provided
answers. You’re allowed to copy my answers, but if you do, make sure to understand what they
mean, and why they are written the way they are. I occasionally include more than one answer,
to give you an idea of what different sorts of things are ok. Things that are equivalent to the
original definition are acceptable answers, but you should make sure that your definitions only
use words that you have already defined. You only need one answer per question.
Vocab Homework Partial Solutions
1) a) A vector space V is a set of things called “vectors”‘ together with a rule for adding
two vectors to get another vector, and a rule for multiplying a vector by a real number to get
another vector.
b) For V to be a vector space, it needs to satisfy the collection of 9 axioms that we were
promised we would not have to memorize!
2) a) A subspace W , of a vector space V , is a subset of the vectors in V .
b) For W to be a subspace of V , it must be true that:
1) The zero vector from V is in W .
2) Given any real number c, and any vector v in W , the vector cv is also in W .
3) The sum of any two vectors in W is also in W .
Alternative answer:
2) a) A subspace W , of a vector space V , is a subset of the vectors in V .
b) For W to be a subspace of V , it must be true that if we equip W with the addition rule
from V , and also the scalar multiplication rule from V , then W is a vector space.
Alternative answer:
2) a) A subspace W , of a vector space V , is a kind of vector space.
b) For W to be a subspace of V , the vectors in W must be a subset of the vectors in V ,
and the addition and scalar multiplication rules in W must be the same rules as those used in V .
3) a) If S is a set of vectors in a vector space V , then a linear combination of S is a vector
in V .
b) For v to be a linear combination of S, there must be vectors v1 , v2 , . . . , vn in S and real
numbers a1 , a2 , . . . , an such that v = a1 v1 + a2 v2 + · · · + an vn .
5) a) If S is a set of vectors in a vector space V , then the span of S is a subspace of V .
b) The span of S is the set of linear combinations of S.
6) a) If S is a set of vectors in a vector space V , then it is possible for S to span V .
b) S spans V if the span of S is V .
Alternative answer:
6) a) If S is a set of vectors in a vector space V , and if W is a subspace of V , then it is possible
for S to span W .
b) S spans W if Span(S) = W .
Alternative answer:
6) a) If S is a set of vectors in a vector space V , and if W is a subspace of V , then it is possible
for S to span W .
b) S spans W if every vector in S is in W , and also if every vector in W is a linear combination of vectors in S.
8) a) A matrix is a collection of numbers arranged into columns and rows in a box.
b) If every column has exactly one number in every row of that column, then those numbers
in a box form a matrix.
10) a) A matrix can have a column space. The column space of a matrix is a vector space.
b) If A is an m × n matrix, then Col(A) is the subspace of Rm consisting of linear combinations of the columns of A.
Alternative answer:
10) a) The column space of a matrix is a vector space.
b) Col(A) is the span of the columns of A.
Alternative answer:
10) a) The column space of an m × n matrix is a subspace of Rm .
b) Col(A) is the set of vectors b in Rm such that the equation Ax = b has a solution.