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.

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