1. Systems of Linear Equations
in Vector Form
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. Any system of linear equations can be put into matrix form:
Or vector form:
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
Here are a few examples:
This is a HOMOGENEOUS system because the right side is all 0.
This is a NON-HOMOGENEOUS system because the right side is not all 0.
This is a NON-HOMOGENEOUS system because the right side is not all 0. Here we have fewer equations than unknowns.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
The lack of a pivot in column three indicates a free variable, and an infinite # of solutions
Row reduction yields:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
The lack of a pivot in column three indicates a free variable, and an infinite # of solutions
Row reduction yields:
Rename X3, call it ‘t’, and we get a vector solution (this is a 1-dimensional subset of ℝ3):
Here is the corresponding solution.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. a2 a1 a3 Here are the vectors for this system.
Because we got a free variable in our row reduction process, we conclude that vectors a1, a2 and a3 are linearly dependent.
Furthermore, since we got 2 pivots in our reduced matrix, we can say that these 3 vectors span a 2-dimensional subset of ℝ3 (a plane, pictured below). a2
This plane will also be called the Column Space of matrix A.
It is also the Span of the set a1 a3
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. Here are the vectors for this system.
Because we got a free variable in our row reduction process, we have infinitely many solutions to the system. The set of all solutions form a 1-dimensional subspace of ℝ3. Since this system is homogeneous, we call this solution set the Null Space of matrix A.
The solution was written as a vector. The Null Space consists of all multiples of this vector. Geometrically, this space is a line in ℝ3, pictured below.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

9. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
Row reduction yields:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

10. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
Row reduction yields:
Here is the unique solution:
This solution tells us the specific linear combination of a1 a2 and a3 that adds up to the right side vector b.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

11. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

12. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
x4 and x5 are free variables
Row reduction yields:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

13. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
x4 and x5 are free variables
Row reduction yields:
Here is the corresponding solution.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

14. Here are the vectors for this system.
The vectors in the vector form of the equation are the columns of the matrix A in the matrix format.
x4 and x5 are free variables
Row reduction yields:
Here is the corresponding solution.
There are 2 free variables, so we get a 2-dimensional subset of ℝ5.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB