Systems of Linear Equations
What is a system of linear equations?
Linear algebra has numerous applications in many diverse fields such as economics, computer science, and engineering. The main reason is that linear models are commonly used in such fields - many real-life problems can be formulated as systems of linear equations. In general, a system of linear equations consists of variables and equations as follows:
,
where are real numbers.
Each equation in the system is a linear equation - an equation that can be written as equality between a real constant and a sum of terms such that each term is a product of a real number and a variable. Real numbers attached to the variables are called coefficients. Usually, the main goal is to find one or more solutions (the values of ) that satisfy this system. To have a deeper understanding of linear systems, we can first interpret a system of linear equations in terms of vectors. In fact, we can rewrite it more elegantly as follows:
It looks familiar, right? The left-hand side is just the linear combination of column vectors formed by the coefficients of the system with those variables as weights. And the system will have a solution if the column vector formed by on the right-hand side is in the span of the column vectors on the left-hand side. Solving the system is equivalent to finding the weights for such a linear combination. We usually call it a vector equation.
Another even more fruitful interpretation of linear systems is through the use of linear transformations.
The Matrix of a Linear Transformation
In the previous section, we already learned that any linear transformation can be uniquely determined by the column vectors in . Now, let us write them down explicitly as column vectors:
We can put those column vectors together in order to form a m x n matrix :
This is the matrix of the linear transformation .
Suppose is an unknown vector in . We write it as a column vector with variable entries: . In other words, . Hence . Explicitly, can be computed by "mulitplying" the matrix of to the column vector as follows:
This linear combination of column vectors can be regarded as the definition of the multiplication of an m x n matrix to an n x 1 matrix.
Suppose is a vector in that can be written as a column vector with as entries i.e. . Then the system of linear equations can be concisely expressed as the matrix equation: . Hence solving the system is equivalent to finding a vector in such that it is transformed to the given vector in by the linear transformation corressponding to the matrix .
Now let us consider the following simple example:
We extract the coefficients of these two linear equations in the system to form a 2 x 2 matrix:
This is the matrix of the linear transformation such that and . Let and . The system can be written as , or more explicitly,
The following are some questions that test your understanding of the concepts you have just learned:
Evaluate .
Suppose the matrix for the linear transformation is . Find a vector in such that . (Note: 0 is the zero vector in .)
Suppose is a linear transformation such that for all in . Find the matrix for .
Suppose is a linear transformation. Find the general form of .
Which of the following statement(s) is/are true? You can select more than one statement.