Linear Algebra : Linear Independence and Rank
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Example Questions
Example Question #11 : Linear Independence And Rank
Consider the following set of three vectors:
where
Is the set linearly independent?
Not enough information
No
Yes
No
Since 
Example Question #12 : Linear Independence And Rank
Does the following row reduced echelon form of a matrix represent a linearly independent set?
Yes
No
Not enough information
No
The set is linearly dependent because there is a row of all zeros.
Notice that having columns of all zeros does not tell if the set is linearly independent or not.
Example Question #13 : Linear Independence And Rank
Determine the row rank of the matrix
To determine the matrix, we turn the matrix into reduced row echelon form
By adding 
And find that the row rank is
Example Question #201 : Operations And Properties
Determine the row rank of the matrix
To determine the row rank of the matrix we reduce the matrix into reduced echelon form.
First we add 
add 
Switch the 2nd row and the 3rd row
multiply the 2nd row by
add 
And we find that the row rank is
Example Question #201 : Operations And Properties
Consider the following set of vectors
Is the the set linearly independent?
Not enough information
No.
Yes.
Yes.
Yes, the set is linearly independent. There are multiple ways to see this
Way 1) Put the vectors into matrix form,
The matrix is already in reduced echelon form. Notice there are three rows that have a nonzero number in them and we started with 3 vectors. Thus the set is linearly independent.
Way 2) Consider the equation
If when we solve the equation, we get 
Distribute the scalar constants to get
Thus we get a system of 3 equations
Since 
Example Question #201 : Operations And Properties
Consider the following set of vectors
Is the the set linearly independent?
No
Not enough information
Yes
No
The vectors have dimension 3. Therefore the largest possible size for a linearly independent set is 3. But there are 4 vectors given. Thus, the set cannot be linearly independent and must be linearly dependent
Another way to see this is by noticing that 
Example Question #202 : Operations And Properties
In a vector space with dimension 5, what is the maximum number of vectors that can be in a linearly independent set?
Five
Not enough information
There is no limit
Ten
Two
Five
The dimension of a vector space is the maximum number of vectors possible in a linearly independent set. (notice you can have linearly independent sets with 5 or less, but never more than 5)
Example Question #203 : Operations And Properties
What is the dimension of the space spanned by the following vectors:
Six
Not enough information
One
Three
Five
Three
Since there are three linearly independent vectors, they span a 3 dimensional space.
Notice that the vectors each have 5 coordinates to them. Therefore they actually span a 3 dimensional subspace of a 5 dimensional space.
Example Question #204 : Operations And Properties
What is the dimension of the space spanned by the following vectors:
Two
One
Three
Not enough information
Five
Three
Since there are three linearly independent vectors, they span a 3 dimensional space.
Notice that the vectors each have 5 coordinates to them. Therefore they actually span a 3 dimensional subspace of a 5 dimensional space.
Example Question #205 : Operations And Properties
True or False: If a 
True
False
True
Since 
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