Linear Systems#

• A linear system is a set of linear equations.
• A linear system is consistent if it has 1 or infinite solutions, and inconsistent if it has no solutions.
• There are only 3 different situations to the solutions of a linear system:
• no solution (inconsistent)
• 1 solution (consistent)
• infinite solutions (consistent)
• A consistent linear system is different from a homogeneous one.
• Consistent means having solutions;
• Homogeneous means a linear system where all the constants are zero.
• Linear systems can be represented by augmented matrices.

REF, RREF, and Elementary Row Operations#

• Augmented matrices (and matrices in general) can be reduced to their Row Echelon and Reduced Row Echelon form via elementary row operations. With the REF and RREF, we can find out the solution, or general solution, to the linear system.
• There are three types of elementary row operations, each corresponding to a type of elementary matrices.
• $R_{i}+kR_{j}$ : add $k$ times the $j^{\text{th}}$ row to the $i^{\text{th}}$ row;
• This approach does not affect the determinant of the matrix;
• This corresponds to the pre-multiplication of the elementary matrix whose $(i, j)$-element is $k$.
• $R_{i}\leftrightarrow R_{j}$ : exchange the $j^{\text{th}}$ row and the $i^{\text{th}}$ row;
• This approach would result in the negation of the determinant of the matrix: if $\textbf{A}\xrightarrow{R_{i}\leftrightarrow R_{j}} \textbf{A}'$, then $\det({\textbf{A}'})=-\det({\textbf{A}})$.
• This corresponds to the pre-multiplication of the elementary matrix whose $i^{\text{th}}$ row and $j^{\text{th}}$ row are exchanged.
• $kR_{i}$: multiply the $i^{\text{th}}$ row by $k$.
• This approach would result in the multiplication of the determinant of the matrix by $k$: if $\textbf{A}\xrightarrow{kR_{i}}\textbf{A}'$, then $\det({\textbf{A}'})=k\det({\textbf{A}})$.
• This corresponds to the pre-multiplication of the elementary matrix whose $(i, i)$-element is $k$ as opposed to $1$.

Invertible Matrices#

• A matrix $\textbf{A}$ is invertible if there exists another matrix $\textbf{B}$ such that $\textbf{A}\textbf{B}=\textbf{I}$ and $\textbf{B}\textbf{A}=\textbf{I}$. $\textbf{B}$ is $\textbf{A}$’s inverse. We denote the inverse of $\textbf{A}$ as $\textbf{A}^{-1}$.
• When a matrix has an inverse, we say that it is invertible. When it has not, we say that it is singular.
• [Important] Let $\textbf{A}$ be a square matrix, and the following properties are equivalent:
• $\textbf{A}$ is an invertible matrix;
• The linear system $\textbf{A}\textbf{x}=\textbf{b}$ has a unique solution;
• The linear system $\textbf{Ax}=\textbf{0}$ has only the trivial solution;
• The Reduced Row Echelon Form of $\textbf{A}$ is $\textbf{I}$;
• $\textbf{A}$ is the product of elementary matrices.
• The determinant of the matrix is not zero. $\det({\textbf{A}})\neq 0$.
• Finding the Inverse
• The inverse of a matrix can be calculated using augmented matrices. With start with $(\textbf{A}|\textbf{I})$, where $\textbf{A}$ and $\textbf{I}$ are square matrices of the same size. Then, we reduce $\textbf{A}$ to $\textbf{I}$ via elementary row operations. Then, $\textbf{I}$ or the right side of our augmented matrix would become $\textbf{A}^{-1}$.
• The inverse of a matrix can also be found with the determinant and the adjugate matrix using the fomula: $$\textbf{A}^{-1}=\dfrac{1}{\det({\textbf{A}})}\text{adj}(\textbf{A})$$
• Adjugate matrix, $\text{adj}(\textbf{A})$, or adjoined matrix, is the transpose of the cofactor matrix. The cofactor matrix is $$\textbf{C}=\left((-1)^{i+j}M_{ij}\right)_{i\times j}$$
• $M_{ij}$, the minor, is the **determinant** of the $(n-1)\times (n-1)$ matrix that removes the $i^{\text{th}}$ row and $j^{\text{th}}$ column.

Determinants#

• The determinant of matrix $\textbf{A}$ is often denoted by $\det({\textbf{A}})$, or $|\textbf{A}|$.
• The determinant for $(a)$ is $a$, the determinant for $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $ac-bd$, and the determinant for $\begin{pmatrix}a & b & c \\ d & e & f \\ g & h & i\end{pmatrix}$ is $aei + bfg + cdh - ceg - afh - bdi$.
• Determinants can be defined recursively.
• To do this, we need to define the minor. Minor, $M_{ij}$, as mentioned above, is the determinant of the $(n-1)\times (n-1)$ matrix that removes $i^{\text{th}}$ row and the $j^{th}$ column. For example, for the matrix $$\begin{pmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{pmatrix}$$ $M_{11}$ would be: $$\begin{vmatrix}a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}$$
• Then, we can find the determinant of a matrix by taking the $i^{\text{th}}$ row, and then use the following formula: $$\det({\textbf{A}})=\sum\limits^{n}{j=1}(-1)^{i+j}M{ij}$$
• Some rules for determinants:
• $\det({\textbf{A}})\det({\textbf{B}})=\det({\textbf{AB}})$
• $\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} + \begin{vmatrix} a'_{11} & a'_{12} & a'_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = \begin{vmatrix} a_{11} + a'_{11} & a_{12} +a'_{12} & a_{13} + a'_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$
• If 1 row / column of the matrix is zero, then the determinant of the matrix is zero.
• If two rows are proportional to each other, than the determinant of the matrix is zero.
• For a diagonal matrix, the determinant of the matrix is the product of all the elements along its main diagonal axis. $\det({\textbf{A}})=\prod\limits^{n}_{i=1}a_{ii}$.
• If the determinant of the matrix $\textbf{A}$ is NOT ZERO, then:
• $\textbf{A}$ is invertible;
• $\textbf{Ax}=\textbf{0}$ has only the trivial solution;
• $\textbf{Ax}=\textbf{b}$ has a unique solution;
• The RREF of $\textbf{A}$ is $\textbf{I}$.

Matrix Calculations#

• Let $\textbf{A}=(a_{ij})_{m\times p}$ and $\textbf{B}=(b_{ij})_{p\times n}$. Then, $\textbf{A}\textbf{B}$ is a $m\times n$ matrix where the $(i,j)$-entry of the matrix is the product of the $i^{\text{th}}$ row of $\textbf{A}$ and the $j^{\text{th}}$ column of $\textbf{B}$. I.e. let $\textbf{AB}=(p_{ij})_{m\times n}$, then \begin{align*}p_{ij}&=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots + a_{ip}b_{pj} \\ &=\sum\limits^{p}_{k=1}a_{ik}b_{kj}\end{align*}
• Matrix Arithmetics ($\textbf{A}$, $\textbf{B}$, $\textbf{C}$ are square matrices of the same size)
• $\textbf{A}+\textbf{B} = \textbf{B} + \textbf{A}$;
• $\textbf{A}\cdot \textbf{B} \neq \textbf{B}\cdot\textbf{A}$ (In most cases);
• $\textbf{A}(\textbf{B} + \textbf{C}) = \textbf{A}\textbf{B} + \textbf{A}\textbf{C}$;
• $(\textbf{A}+\textbf{B})^{2}=\textbf{A}^{2}+\textbf{AB}+\textbf{BA}+\textbf{B}^{2}$;
• $\textbf{A}\textbf{I}=\textbf{I}\textbf{A}=\textbf{A}$;
• $\textbf{A}\textbf{0}=\textbf{0}\textbf{A}=\textbf{0}$.

Spans and Subspaces#

• What does $\text{span}(v_{1}, v_{2}, \cdots, v_{n})$ mean?
• It means the set which contains all linear combinations of $v_{1}, v_{2}, \cdots, v_{n}$
• How to pove that $\text{span}(v_{1}, v_{2}, \cdots, v_{n})\subseteq \text{span}(u_{1}, u_{2}, \cdots, u_{m})$ ?
• We can create the augmented matrix $\begin{pmatrix} u_{1} & u_{2} & \cdots & u_{m} & | & v_{1} & v_{2} & \cdots & v_{n}\end{pmatrix}$, then reduce the left part to REF to find whether if the system if consistent.
• If the system is consistent, then $\text{span}(v_{1}, v_{2}, \cdots, v_{n})\subseteq \text{span}(u_{1}, u_{2}, \cdots, u_{m})$ is true. Otherwise, this is not true.
• If we were to prove two spans are equal, then we prove $\text{span1}\subseteq \text{span2}$ and $\text{span2}\subseteq \text{span1}$.
• A set $S$ is a subspace of $\mathbb{R}^{n}$ if $S=\text{span}(v_{1}, v_{2}, \cdots, v_{n})$ where $v_{1}, v_{2}, \cdots, v_{n}\in \mathbb{R}^{n}$.
• What are the three conditions to satisfy for a set to be a subspace?
• It is not empty: $\textbf{0} \in S$;
• Addition is contained: if $\textbf{u}, \textbf{v} \in S$, then $\textbf{u} + \textbf{v} \in S$;
• Scalar multiplication is contained: if $\textbf{u}\in S$, then $k\textbf{u}\in S, k \in \mathbb{R}$.
• Note that the solution set of a homogeneous system is a subspace because it satisfies that:
• $\textbf{0}\in S$
• If $\textbf{x}_{0}$ is a solution to the homogeneous system $\textbf{Ax}=\textbf{0}$, then $\textbf{A}(k\textbf{x}_{0})=k\textbf{0}=\textbf{0}$, then $x_{0}'=k\textbf{x}_{0}$ is also a solution.
• If both $\textbf{x}_{0}$ and $\textbf{x}_{1}$ are solutions to the homogeneous system $\textbf{A}\textbf{x}=\textbf{0}$, then $\textbf{A}(\textbf{x}_{0}+\textbf{x}_{1})=\textbf{0}+\textbf{0}=\textbf{0}$, therefore $\textbf{x}_{2}=\textbf{x}_{0} + \textbf{x}_{1}$ is also a solution.
• What is the alternative definition for subspace? (Used in abstract linear algebra)
• Let $V$ be an non-empty subset of $\mathbb{R}^{n}$, then $V$ is a subspace of $\mathbb{R}^{n}$ if and only if $$(\forall \textbf{u}, \textbf{v} \in V)(\forall c, d\in R)[c\textbf{u} + d\textbf{v} \in V]$$

Change Log#

• Fixed issue where I mistyped “not zero” as “zero” right before the matrix calculations section;
• Fixed issue where I mistyped the formula \begin{align*}p_{ij}&=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots + a_{ip}b_{pj} \ &=\sum\limits^{p}_{k=1}a_{ik}b_{kj}\end{align*}
• Fixed the issue where I mistyped homogeneous as homologous;
• Fixed the issue where I said that $\textbf{A}$ is invertible if $\textbf{AB}=\textbf{I}$, which shall be true for square matrices and is sufficient to prove invertibility. But for the sake of being more accurate, it should be that $\textbf{A}$ is invertible iff $\textbf{AB}=I$ and $\textbf{BA}=\textbf{I}$;
• Fixed the issue where I mistyped the formula for finding the determinant for triangular matrices, where I typed $\det({\textbf{A}})=\sum\limits^{n}_{i=1}a_{ii}$ as opposed to $\det({\textbf{A}})=\prod\limits^{n}_{i=1}a_{ii}$;
• Fixed the issue where I miss typed the determinant expansion for the 3 by 3 matrix.

If you find more typos and small mistakes, please feel free to email me.