notes-linear-algebra-with-gilbert-strang/notes.org

Abreviations

• RHS: Right hand side

Lecture 1

Example 1

Equations

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

Parts

Coefficients

[ 2 -1 ] [ -1 2 ]

Unknowns

[ x ] [ y ]

Right hand sides

[ 0 ] [ 3 ]

Matrix form

Written as matrix form:

[ 2 -1 ] [ x ] = [ 0 ] [ -1 2 ] [ y ] = [ 3 ]

Usually will name the parts:

A X = b

Row picture

• draw a line in a coordinate system per row
• look at where the lines intersect
• read the coordinates at the intersection point -> that is the solution to the system of equations

Column picture

[ 2] [-1] [ 0] x [ ] + y [ ] = [ ] [-1] [ 2] [ 3]

Seeing the columns as vectors.

How to combine the vectors to find the right amount (right hand side)? -> Linear combination! Most important operation in the course!

Find numbers x and y (the linear combination) to get the right amount (right hand side).

• draw the vectors in a coordinate system
• see how many times one needs to step in the direction of the vectors to get to the right hand side coordinates
• the times are the result for x and y

Example 2

• 3 equations
• 3 unknowns

+2x -1y = 0 -1x +2y -1z = -1 -3y +4z = 4

[+2 -1 0] [ 0] A = [-1 +2 -1] b = [-1] [ 0 -3 +4] [+4]

Row picture

• 3d coordinate system
• all points that solve an equation will result in a plane of points
• intersection of 2 planes: usually a line, unless planes are equal
• intersection of 3 planes: 2 was a line, a line intersects with plane usually in one point, unless the line is on the plane
• -> there is a point where 3 planes intersect, except for special cases
• parallel planes
• equal planes (no new information!)
• difficult to see in a graphic where the intersection point it.
• quit the row picture!

Column picture

[ 2] [-1] [ 0] [ 0] x [-1] + y [-2] + z [-1] = [-1] [ 0] [-3] [ 4] [ 4]

• What combination of those 3 vectors will get the right hand side?
• draw vectors in 3d coordinate system
• how many times of each vector we need to get to the RHS point?
• the RHS is already at 1z!
• no need to take the other vectors!
• -> 0x + 0y + 1z = b

Can we solve Ax = b for every b?

• Idea: As long as we have some amount of each of the components x y z in the coordinate system, I think there should be a solution for every b.
• Only partially correct.
• If the 3 columns of the matrix lie in the same plane, they cannot solve for every possible b!
• -> We might need to check, whether the columns are in the same plane.
• Vectors are in the same plane, if ???
• The vectors need to be independent from each other.
• What does that mean?
• What does Ax mean?
• Trying to find a linear combination, a vector, which will result in the RHS.

Matix form

Ax = b

with:

[+2 -1 0] [ 0] A = [-1 +2 -1] b = [-1] [ 0 -3 +4] [+4]

How to multiply a matrix by a vector?

A·x = ?

Column wise:

"one time the first column, two times the second column"

[ 2 5] [ 1] [ 2] [ 5] [12] [ ] · [ ] = 1 [ ] + 2 [ ] = [ ] [ 1 3] [ 2] [ 1] [ 3] [ 7]

Or row wise (dot product view):

[ 2 5] [ 1] [1*2 + 2*5] [12] [ ] · [ ] = [ ] = [ ] [ 1 3] [ 2] [1*1 + 2*3] [ 7]

Lecture 2

TODO