7- Vector Spaces, Span, Basis, Null Space

Question 1

Regarding the determinant of a matrix A:
- its absolute value represents…
- its sign tells us…
- if it is zero then it means that…

Answer 1

• how much the transformation scales the areas or volumes.
• whether the transformation preserves or reverses the orientation.
• if it’s zero then the transformation is not invertible and it collapses the space.

Question 2

• What is the vector space R^n
• What is a vector subspace?

Answer 2

• set of all vectors v with n components that are real numbers.
• A subspace H of a vector space V is a subset of vectors of V with 3 properties:
  1. it is closed under the vector addition;
  2. it is closed under the scalar multiplication;
  3. it contains the zero vector.

Question 3

What is the span of n vectors?
- What is the span of the empty set?
- What is the span of a single non zero vector?
- WHat is the span of 2 linearly independent vectors?
- WHat is the span of 3 linearly independent vectors?

Is the span of k vectors always k dimensional?
How many dimension is the span of the zero vector?

Answer 3

• Set of all their linear combination: all the possible vectors that you can reach using only scalar multiplication and vector addition.
• just the origin
• it is the line through the origin and v: {alfa*v | alfa belongs to R}
• 2d space
• 3d space.

-No, it depends on whether there are linearly independent vectors or not.
- zero

Question 4

A set of vectors is…
- linearly independent (definition and notation)
- linearly dependent (definition and notation)

Answer 4

• independent if there are not vectors that can be expressed as linear combination of the others: if x1v1+…+xnvn = 0 has the only solution x= 0.
• dependent if there is at least 1 vector which can be described as linear combination of the others: if x1v1+…+xnvn = 0 has at list a solution x!= 0.

Question 5

• the columns of a matrix are linearly independent if…
• if a set contains more vectors than there are entries in each vector… and so column vectors of a matrix A…
• if a set contains the zero vector then…
• what is a basis?

Answer 5

• if Ax = 0 has the only solution x= 0;
• the set is linearly dependent: given Amxn, if n>m then the column vectors are linearly dependent.
• the set is linearly dependent.
• it is the set of linearly independent vectors that span the full space.

Question 6

Each matrix A has 2 vector spaces associated with itself…

How can we define a basis for the column space?
How can we define a basis for the row space?

Answer 6

• the column space of A= span(cols of A)
• the row space of A= span(rows of A)
• basis col space = pivot columns of matrix A;
• basis row space = non zero row of its echelon form.

Question 7

Null Space of a Matrix A:
- it is the set of…
- notation
- it is a subspace of…
- how to find a basis for the nut space

Answer 7

• set of all the solutions x such that Ax=0
• NulA= {x: x∈R^n and Ax=0}
• subspace of R^n
• we find the solutions of Ax=0 and then we take the vectors associated with the free variables.

Question 8

What is the dimension dimH of a non-zero subspace H?
What is the dimension of a zero subspace {0}?
What is the dimension of NulA?

Answer 8

• number of vectors in any basis for H
• 0
• number of free variables.

Question 9

What does the rank of a matrix represent?
It is equal to… that is equal to…

Answer 9

• how much info a matrix carries
• is the dimension of the column space that is equal to the number of pivot columns of the matrix.