**System of Linear Equation, Row Reduction and Echelon Form**

- Three row operations:
**replacement**(add multiple of a row to another),**interchange**,**scaling**(multiply a row by a non-zero number). - Example of an
**echelon form**\begin{bmatrix}* & * & * & * \\ 0 & 0 & * & * \\ 0 & 0 & 0 & *\end{bmatrix}. - Example of a
**reduced echelon form**\begin{bmatrix}1 & * & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}. - Question: Does every entry in a reduced echelon form have to be either 0 or 1?
**Pivot position**: the positions of the leading entries in the echelon form.**Pivot column**: the columns of the pivot positions.- 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**. **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**

- It only makes sense to multiply a matrix with a vector if the number of columns is equal to the dimension of the vector.
**Row vector rule**: use dot product to compute multiplication of a matrix with a vector.- 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**

**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}.- 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}. - Question: Is homogeneous system always consistent?

**Linear Independence**

- 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. - If p > n, then the set of n dimensional vectors \{v_1, \ldots, v_p\} is always
**linearly dependent.** **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.- 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**

- A linear transformation T is associated to its
**standard matrix**A in the sense that T(x)=Ax for all vectors x. - 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. - 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.