Many topics in linear algebra are deeply interconnected. A result in the context of one topic is often expressible in the context of another topic. The Big Theorem, below, denomstrates just how true this is.
Recall that two propositions \(P\) and \(Q\) are equivalent if \(P\Rightarrow Q\) and \(Q\Rightarrow P\) (or, more succinctly, \(P\Leftrightarrow Q\)).
The Big Theorem shows that the invertibility of a given \(n\times n\) matrix is equivalent to dozens of other properties in various contexts: solutions sets to \(A\vec{x}=\vec{b}\) and \(A\vec{x}=\vec{0}\text{,}\) linear combinations of rows and columns of \(A\text{,}\) row-echelon form, pivots, determinants, linear independence, span, bases and dimension, column and row spaces and nullspaces, similarity, projections, eigenvalues, and linear transformations\(A.\)
TheoremA.5.1.The Big Theorem.
Let \(A\in\R^{n\times n}.\) The following are equivalent:
\(\A\) is invertible.
\(\A^T\) is invertible.
\(\A\x=\b\) has a solution for every \(\b\in\R^n.\)
\(\A^T\x=\b\) has a solution for every \(\b\in\R^n.\)
For a given \(\b\in\R^n,\) if \(\A\x=\b\) has a solution that solution is unique.
For a given \(\b\in\R^n,\) if \(\A^T\x=\b\) has a solution that solution is unique.
\(\A\x=\vec{0}\) has the unique solution \(\x=\vec{0}.\) (Theorem 3.10.3)
\(\A^T\x=\vec{0}\) has the unique solution \(\x=\vec{0}.\)
\(A\in\R^{n\times n}\) is singular if and only if there exist distinct \(\u,\v\in\R^n\) for which \(\A\u=\A\v.\) (Theorem 5.4.9)
\(A^T\in\R^{n\times n}\) is singular if and only if there exist distinct \(\u,\v\in\R^n\) for which \(\A^T\u=\A^T\v.\)
The zero vector \((0,0,\ldots,0)\) is not a nontrivial linear combination of the rows of \(\A.\)
The zero vector \((0,0,\ldots,0)^T\) is not a nontrivial linear combination of the columns of \(\A.\)
\(\A\) is row-equivalent to \(\I.\)
\(\A^T\) is row-equivalent to \(\I.\)
No row of \(\A\) is a linear combination of the other rows of \(\A.\)
No row of \(\A^T\) is a linear combination of the other rows of \(\A^T.\)
No column of \(\A\) is a linear combination of the other columns of \(\A.\)
No column of \(\A^T\) is a linear combination of the other columns of \(\A^T.\)
Every row-echelon form \(\U\) of \(\A\) is invertible.
Every row-echelon form \(\U\) of \(\A\) shows a full set of pivots.
The row-echelon form of \(\A^T\) shows a full set of pivots.
Every row-echelon form \(\U\) of \(\A\) has no zero rows.
Every row-echelon form \(\U\) of \(\A^T\) has no zero rows.
The reduced row-echelon form \(\J\) of \(\A\) is the identity.
The reduced row-echelon form \(\J\) of \(\A^T\) is the identity.
\(\displaystyle \det(\A)\ne0.\)
\(\displaystyle \det(\A^T)\ne0.\)
The set of columns of \(\A\) is linearly independent.
The set of columns of \(\A^T\) is linearly independent.
The set of rows of \(\A\) is linearly independent.
The set of rows of \(\A^T\) is linearly independent.
The set of columns of \(\A\) spans \(\R^n.\)
The set of columns of \(\A^T\) spans \(\R^n.\)
The set of rows of \(\A\) spans \(\R^n.\)
The set of rows of \(\A^T\) spans \(\R^n.\)
The set of columns of \(\A\) is a basis for \(\R^n.\)
The set of rows of \(\A\) is a basis for \(\R^n.\)
The set of columns of \(\A^T\) is a basis for \(\R^n.\)
The set of rows of \(\A^T\) is a basis for \(\R^n.\)
\(\displaystyle r(\A)=n.\)
\(\displaystyle r(\A^T)=n.\)
col\((\A)=\R^n.\)
col\((\A^T)=\R^n.\)
row\((\A)=\R^n.\)
row\((\A^T)=\R^n.\)
dim(col\((\A))=n.\)
dim(col\((\A^T))=n.\)
dim(row\((\A))=n.\)
dim(row\((\A^T))=n.\)
\(\displaystyle N(\A)=\left\{\vec{0}\right\}.\)
\(\displaystyle N(\A^T)=\left\{\vec{0}\right\}.\)
dim\((N(\A))=0.\)
dim\((N(\A^T))=0.\)
There is no basis for \(N(\A).\)
There is no basis for \(N(\A^T).\)
\(\A\) is similar to some invertible matrix.
\(\A^T\) is similar to some invertible matrix.
\(\A\) is similar to \(\I.\)
\(\A^T\) is similar to \(\I.\)
The projection matrix \(\P\) to col\((\A)\) is the identity.
The projection matrix \(\P\) to col\((\A^T)\) is the identity.
No eigenvalue of \(\A\) is zero.
No eigenvalue of \(\A^T\) is zero.
The linear transformation \(T_\A\) associated with \(\A\) is injective.
The linear transformation \(T_\A\) associated with \(\A\) is surjective.
The linear transformation \(T_{\A^T}\) associated with \(\A^T\) is injective.
The linear transformation \(T_{\A^T}\) associated with \(\A^T\) is surjective.
