Quiz 3

Question 1

What is a spanning set of vectors for a vector space V?

Answer 1

A spanning set is any set of vectors which can span V

Question 2

What is a basis for a vector space V?

Answer 2

A basis is a linearly independent set of vectors which can span V

Question 3

If the set S = {v1,v2,…,vn} is a basis for a vector space V, what can be said about all linearly independent sets of vectors in V?

Answer 3

They must be n or smaller.

Question 4

What can be said about the dimension of a vector space?

Answer 4

It is equal to the number of vectors required for a basis of that vector space

Question 5

What order should you go about proving a set of vectors is a basis?

Answer 5

First show it spans, then plug in 0 to show it is linearly independent

Question 6

How do you write the coordinates with respect to a certain basis?

Answer 6

Find the coefficients which would result in the vectors of that basis equaling the coordinates, then write those coefficient into a nx1 matrix (aka a n-tuple)

Question 7

What is the row space of a matrix?

Answer 7

The row space of a matrix is the vector space spanned by the rows of that matrix

Question 8

What is the row rank of a matrix and how do you find it?

Answer 8

The row rank is the dimension of the row space, to find it, simply row reduce the matrix and count the number of non-zero rows.

Question 9

What is the basis for the row space?

Answer 9

It is the set of non-zero rows which remain after the matrix has been row-reduced using only elementary row operations

Question 10

What is true of the column rank of a matrix?

Answer 10

It is the same as the row rank

Question 11

How do you denote the row rank of a matrix A?

Answer 11

r(A)

Question 12

What does a system of equations being consistent and r(A) = k imply?

Answer 12

That the solutions to the system can be expressed using n-k arbitrary unknowns where n denotes the total number of unknowns in the system.

Importantly, if n-k is 0, that means there is a unique solution to the system of equations

Question 13

What are the conditions for a system of a equations to be consistent?

Answer 13

For the system Ax=b it is consistent if and only if r(A) = r(A|b)

Essentially if you add the column b to the matrix and row reduce, is the rank greater than if b was not present.

Question 14

What is the rank of a matrix?

Answer 14

It is its column rank, which is its row rank

Question 15

How can you use row rank to determine linear independence?

Answer 15

Take a set of k n-tuples. If the row rank of the resulting matrix is not equal to k then the set is not linearly independent.

Question 16

What does rank tell us about invertibility?

Answer 16

An nxn matrix has an inverse if and only if its rank is equal to n