Linear Algebra : Linear Algebra
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Example Questions
Example Question #1 : Linear Independence And Rank
If matrix A is a 10x12 matrix with a three-dimensional null space, what is the rank of A?
None of the other answers
Given that rank A + dimensional null space of A = total number of columns, we can determine rank A = total number of columns-dimensional null space of A. Using the information given in the question we can solve for rank A:
Example Question #1 : Linear Independence And Rank
Does the following row reduced echelon form of a matrix represent a linearly independent set?
Yes
No
Not enough information
Yes
The set must be linearly independent because there are no rows of all zeros. There are columns of all zeros, but columns do not tell us if the set is linearly independent or not.
Example Question #1 : Linear Independence And Rank
In a vector space of dimension 5, can you have a linearly independent set of 3 vectors?
Yes
No
Not enough information
Yes
The dimension of the vector space is the maximum number of vectors in a linearly independent set. It is possible to have linearly independent sets with less vectors than the dimension.
So for this example it is possible to have linear independent sets with
1 vector, or 2 vectors, or 3 vectors, all the way up to 5 vectors.
Example Question #1 : Linear Independence And Rank
Consider a set of 3 vectors from a 3 dimensional vector space.
Is the set linearly independent?
Not enough information
No
Yes
Not enough information
It depends on what the vectors are.
For example, if
Then the set is linearly independent.
However if the vectors were
then the set would be linearly dependent.
Example Question #1 : Linear Independence And Rank
Consider a set of 3 vectors from a 2 dimensional vector space.
Is the set linearly independent?
Yes
Not enough information
No
No
Since the dimension of the space is 2, a linearly independent set can have at most two vectors. Since the set in consideration has 3 and 3>2, the set must be linearly dependent.
Example Question #191 : Operations And Properties
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?
No
Not enough information
Yes
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 #192 : Operations And Properties
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 
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