Linear Algebra Lecture 1

Linear Algebra Lecture 1

1. n linear equations & n unknowns

2. Row picture

3. Column picture

4. Matrix form

The fundamental problem of linear algebra which is to solve a system of linear equations.

Example 1:

{2x−y=0−x+2y=3

In matrix form :

[2−1−12][xy] = [03]

The matrix A, means the matrix of coefficients (系数矩阵).

The vector of unknowns marks as X.

The right-hand side is also a vector called b.

So linear equations are Ax = b.

The row picture of the example :

take one row at a time and draw the XY plane.

Plot all the points that satisfy those equations.

The column picture of the example :

x[2−1]+y[−12]=[03]

The equation is asking us to find to combine the vectors, to find right linear combination(线性组合).

Linear combination of columns is the most fundamental operation in the whole course.

The right combination going to take is the right one to produce (0,3).

1[2−1]+2[−12]=[03]

Example 2:

⎧⎩⎨2x−y=0−x+2y−z=−1−3y+4z=4

A=⎡⎣⎢2−10−12−30−14⎤⎦⎥, b=⎡⎣⎢0−14⎤⎦⎥

The row picture is three planes, they’re not parallel and not special. They do meet in one point and that’s the solution.

The row picture was a cinch when we looked at two lines meeting. When we look at three planes meeting, it’s not so clear and in four dimensions probably a little less clear. So quit on the row picture and take the column picture.

The column picture of the example:

x⎡⎣⎢2−10⎤⎦⎥+y⎡⎣⎢−12−3⎤⎦⎥+z⎡⎣⎢0−14⎤⎦⎥=⎡⎣⎢0−14⎤⎦⎥

The left-hand side is a linear combination of three vectors, each one is a three dimensional vector.

Col3=b

so x=0,y=0,z=1

Elimination

Elimination is the systematic way that everybody, every bit of software, production, large-scale software would solve the equations.

Can I solve Ax=b for every b ? NO

In linear combination words is :

Do the linear combinations of the columns fill three dimensional space?

Every b means all the **b**s in three dimensional space.

A times x, when multiply a matrix by a vector, get a combination of column.

For the matrix A in example 2, the answer is YES, it’s a non-singular matrix(非奇异矩阵), an invertible matrix(可逆矩阵).

The other answer is NO, when would I not be able to produce some b?

If these three columns all lie in the same plane, then their combinations will lie in that same plane. Only right-hand side b that I could get would be the ones in that plane, so could solve it for some right-hand sides, when b is in the plane, but most right-hand sides would be out of the plane and unreachable. So that would be a singular matrix(奇异矩阵), the matrix would be not invertible, So there would not be a solution for every b.

Ax is the combinations of columns of A.