Recitation 5A

System of Linear Equation, Row Reduction and Echelon Form

1. Three row operations: replacement (add multiple of a row to another), interchange, scaling (multiply a row by a non-zero number).
2. Example of an echelon form $\begin{bmatrix}* & * & * & * \\ 0 & 0 & * & * \\ 0 & 0 & 0 & *\end{bmatrix}.$
3. Example of a reduced echelon form $\begin{bmatrix}1 & * & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}.$
4. Question: Does every entry in a reduced echelon form have to be either 0 or 1?
5. Pivot position: the positions of the leading entries in the echelon form.
6. Pivot column: the columns of the pivot positions.
7. Given a system of linear equations, the variables correspondent to pivot columns of the augmented matrix are basic variables. The rest of the variables are free variables.
8. Existence and uniqueness theorem: A system of linear equations is consistent if and only if the rightmost column of its augmented matrix is not a pivot column. A consistent system of linear equations has a unique solution if and only if it has no free variables.

Vector Equation and Matrix Equation

1. It only makes sense to multiply a matrix with a vector if the number of columns is equal to the dimension of the vector.
2. Row vector rule: use dot product to compute multiplication of a matrix with a vector.
3. Question: Given a constant $c$ and a vector $x\in\mathbb{R}^n$. Find out a matrix $A$ such that $c\mathbf{x}=A\mathbf{x}$ for all $\mathbf{x}$.

Solution Sets of Linear Systems

1. Parametric vector form: One can write the solution set of a linear system in the vector form such as $\mathbf{x}=\mathbf{p}+s\mathbf{u}+t\mathbf{v}$. The geometric interpretation of this specific solution set is a plane through the point $\mathbf{p}$ spanned by $\mathbf{u}, \mathbf{v}$.
2. Suppose the solution set of the homogeneous system $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=s\mathbf{u}+t\mathbf{v}$ and the non-homogeneous system $A\mathbf{x}=\mathbf{b}$ has a solution $\mathbf{p}$. Then the solution set of the non-homogeneous system is exactly $\mathbf{x}=\mathbf{p}+s\mathbf{u}+t\mathbf{v}$, the solution set of the homogeneous one translated by $\mathbf{p}$.
3. Question: Is homogeneous system always consistent?

Linear Independence

1. A set of $n$ dimensional vectors $\{v_1, \ldots, v_p\}$ is linearly independent if and only if $v_1x_1 + \ldots + v_px_p=0$ has only the trivial solution.
2. If $p > n$, then the set of $n$ dimensional vectors $\{v_1, \ldots, v_p\}$ is always linearly dependent.
3. Characterization of Linear Dependence: the set of vectors $\{v_1, \ldots, v_p\}$ is linearly dependent if and only if one of the vectors is a linear combination of the rest.
4. Question: Suppose $\{v_1, v_2, v_3\}$ is linearly dependent. Is it necessarily true that $v_1$ is a linear combination of $v_2, v_3$?

Linear Transformation

1. A linear transformation $T$ is associated to its standard matrix $A$ in the sense that $T(x)=Ax$ for all vectors $x$.
2. A linear transformation $T$ is one-to-one if and only if $Ax=0$ only has the trivial solution, where $A$ is $T$‘s standard matrix. It is onto if and only if $Ax=b$ has a solution for all $b$.
3. Practically, in order to see if $T$ is one-to-one, one only needs to check if all columns of $A$ are pivot columns. To see if $T$ is onto, one only needs to check if every row of $A$ contains a pivot position.