Two linear spaces are isomorphic if and only if they are of the same dimension, or, equivalently, the size of a basis of each includes the same number of elements.

 is the set of  matrices; each such matrix includes 4 elements, so it has four elements in one of its bases; for example, one such basis is

 is the set of all polynomials in  of degree 4 or less - that is, all polynomials of the form

Each polynomial is defined by five coefficients, so the dimension of  is 5. Each basis of  has five elements; for example, one such basis is

.

Since , the spaces are not isomorphic.

  is a vector space with dimension 2, so any basis for  must include exactly two vectors. This makes it possible that  forms a basis for ; also, this makes their linear independence necessary and sufficient for them to form a basis, since, if they are as such, they are a spanning set as well.

We can test for linear independence by forming the following matrix with their entries:

The rows (and the original vectors) are linearly independent if and only if 

,

so calculate the determinant:

Applying the rules of logarithms:

The rows of the matrix, and, consequently, the vectors  and , are linearly dependent. Furthermore,  and  do not span .

The first step, is to create an augmented matrix with the identity Matrix.

To find the inverse, all we need to do is get the Identity Matrix on the left hand side.

Since we have the Identity Matrix on the left hand side, we are done solving for the inverse.

To find the inverse, first find the determinant. In this case, the determinant is

The inverse is found by multiplying

First, find the determinant:

Now multiply by the matrix , the original matrix with 2 and 5 switched and the signs changed on -1 and 0.

For any 2x2 matrix, to determine if it is invertible, we must first calculate its determinant. If the determinant is equal to 0, then the matrix is not invertible. If it isn’t equal to 0, then its inverse can be found using this formula: 

For any 2x2 matrix, to determine if it is invertible, we must first calculate its determinant. If the determinant is equal to 0, then the matrix is not invertible. If it isn’t equal to 0, then its inverse can be found using this formula: 

The matrix is not square so it does not have an inverse. 

The matrix is square, so it could have an inverse.  Next we find the determinant.  This matrix has a determinant of 0, so it does not have an inverse. 

To find the inverse of a matrix first look to verify that the matrix is square.  If it is not square, it does not have an inverse. Next, you must find the determinant.  If the determinant is 0, then the matrix does not have an inverse.  The determinant for this matrix is ad-bc = 9, therefore it has an inverse.  To find the inverse of a 2x2 matrix we first write it in augmented form.

    First we will divide R1/2  next we will eliminate the first column by taking R2-5R1 , next we will divide 9R2/2 to set the second pivot. . Next we will eliminate the second column by taking R1+1/2R2. . Now that we have the identity matrix on the left, our answer is on the right.  There is, however, an easier way to determine the inverse of a 2x2 matrix.  The trick is to swap the numbers in spots a and d, put negatives in front of the numbers in spots b and c and then divide everything by the determinant.  For this example,  then divide by the determinant which is 9 and simplify.