
16.1: Summary

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We have seen that a linear transformation has an inverse if and only if it is bijective ($$\textit{i.e.}$$, one-to-one and onto).  We also know that linear transformations can be represented by matrices, and we have seen many ways to tell whether a matrix is invertible.  Here is a list of them:

Theorem        (Invertibility)

Let $$M$$ be an $$n \times n$$ matrix, and let $$L \colon \Re^{n} \to \Re^{n}$$ be the linear transformation defined by $$L(v)=Mv$$. Then the following statements are equivalent:

a)    If $$V$$ is any vector in $$\Re^{n}$$, then the system $$MX=V$$ has exactly one solution.

b)    The matrix $$M$$ is row-equivalent to the identity matrix.

c)    If $$v$$ is any vector in $$\Re^{n}$$, then $$L(x)=v$$ has exactly one solution.

d)    The matrix $$M$$ is invertible.

e)    The homogeneous system $$MX=0$$ has no non-zero solutions.

f)     The determinant of $$M$$ is not equal to $$0$$.

g)    The matrix $$M$$ is a product of elementary matrices of the form $$E_{j}^{i}, R^{i}(\lambda), S_{j}^{i}(\gamma)$$ with $$\lambda \neq 0$$.

h)    The transpose matrix $$M^{T}$$ is invertible.

i)     The matrix $$M$$ does not have $$0$$ as an eigenvalue.

j)     The linear transformation $$L$$ does not have $$0$$ as an eigenvalue.

k)    The characteristic polynomial $$\det(\lambda I-M)$$ does not have $$0$$ as a root.

l)     The columns (or rows) of $$M$$ span $$\Re^n$$.

m)   The columns (or rows) of $$M$$ are linearly independent.

n)    The columns (or rows) of $$M$$ are a basis for $$\Re^{n}$$.

o)    The linear transformation $$L$$ is injective.

p)    The linear transformation $$L$$ is surjective.

q)    The linear transformation $$L$$ is bijective.

Note: it is important that $$M$$ be an $$n \times n$$ matrix! If $$M$$ is not square, then it can't be invertible, and many of the statements above are no longer equivalent to each other.

Proof
Many of these equivalences were proved earlier in other chapters.  Some were left as review questions or sample final questions.  The rest are left as exercises for the reader.