Linear Algebra--Gilbert Strang

Linear Algebra—Gilbert Strang

First lecture

Linear Equations:

Definition

n linear equations with n unknowns

plan

• Row picture

• Column picture

• Matrix form(algebra way)

examples

A system of Linear Equations:

$$2x - y = 0 \\ -x + 2y = 3$$

Its matrix form:

$$\begin{bmatrix} 2& -1 \\ -1& 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix}$$ $$A \textbf{X} = b$$

$A$: Matrix of coefficients

$X$: Vector of unknowns

$b$: Vector

Row picture:

Solution: The point that lies on both line of the equations.

Column picture(vector picture):

$$x \begin{bmatrix} 2 \\ -1 \end{bmatrix} + y \begin{bmatrix} -1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix}$$ $$Linear\ combination\ of\ the\ columns$$

Question:

How to combine the vectors given in right amounts to get the last one?

It equals to:

Finding the right linear combination of the columns.

What about all the combinations?

The combinations of the two vectors will fill the whole plane.

Question:

Can I solve the equations for every given b?

Or in n linear combination words:

Do the linear combinations of the columns fill the whole space?

Answer:

Yes.

Because the given matrix is a non-singular matrix, or to say: an invertible matrix.

If not:

The matrix is singular, or to say, not invertible.

How to multiply a matrix by a vector?

Given:

$$A = \begin{bmatrix} a_{1, 1}& a_{1, 2}& \cdots& a_{1, m} \\ a_{2, 1}& a_{2, 2}& \cdots& a_{2, m}& \\ \vdots& \vdots& \ddots& \vdots& \\ a_{n, 1}& a_{n, 2}& \cdots& a_{n, m} \end{bmatrix}$$ $$\textbf{X} = \begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_m \end{bmatrix}$$

Calculate:

$$A \textbf{X} = b$$

Answer:

$$b = \begin{bmatrix} \sum _{i = 1} ^m a_{1, i} c_i \\ \sum _{i = 1} ^m a_{2, i} c_i \\ \vdots \\ \sum _{i = 1} ^ m a_{n, i} c_i\end{bmatrix}$$ $$or$$ $$b = \sum _{i = 1} ^ m \begin{bmatrix} a_{1, i} c_i \\ a_{2, i} c_i \\ \vdots \\ a_{n, i} c_i\end{bmatrix}$$

Meaning:

Linear combination of columns.

Second lecture: matrix elimination

Key idea of the course:

matrix operations

Purpose:

reduce the variables

Pivot

The first non-zero element of every row

matrix elimination steps

Step 1

Reduce every elements on the rows below the pivot using times of the row.

Step 2

Repeat the reducing till the matrix becomes a triangular form.

Failure:

1. The first element of the first row is zero.

   Solution: Exchange the row with the rows below.

2. There is not a pivot on the lowest row.

   Conclusion: The matrix is not reversible.