MIT 18.06 by Glibert Strang
§1 The Geometry of Linear Equations
Give the follow equation
{2x−y=0−x+2y=3(1.1)
the matrix form
[2−1−12][xy]=[03](1.2)
equation (1.4) is a Linear Combination
x[2−1]+y[−12]=[30](1.3)
the euqtion (1.3) can be written as a normal form, x is a vector,
Ax=b(1.4)
we are supposed to solve the x while x is not always exits, here's an example
=x⎣⎡2−10⎦⎤+y⎣⎡−120⎦⎤+z⎣⎡0−10⎦⎤⎣⎡0−1−4⎦⎤(1.5)
In this case, column1, column2 and column3 are in the same plane, then their combinations will lie in that same plane. so this would be a singular case, the martix would be not invertible. There would be no a solution for any b.
After, we can get a conclusion that for Ax=b:
Ax is a combination of columns of A
for the follow example,
[2153][12]=1×[21]+2×[53]=[21]+[106]=[127](1.6)
§2 Elimination with Matrices
The key idea of Elimination is Matrix Operation
Give the follow equation
⎩⎪⎨⎪⎧x+2y+z=23x+8y+z=124y+z=2(2.1)
Give the determinant form of the equation
∣∣∣∣∣∣130284111∣∣∣∣∣∣(2.2)
The red 1 is called 1st pivot; then we row2 minus 3×row1, get the follow determinant
∣∣∣∣∣∣1002241−21∣∣∣∣∣∣(2.3)
then row3 minus 2×row2
∣∣∣∣∣∣100220105∣∣∣∣∣∣(2.4)
write the right hand of equation (2.1) into the matrix (2.2), it's called Augmented Matrix
⎣⎡1302841112122⎦⎤(2.5)
do the same elimination for matrix (2.5)
⎣⎡1002201−2526−10⎦⎤(2.6)
The Rule of Matrix Multiplication
Matrix×column=columnrow×Matrix=row
The reasult of multiplying a matrix by some vector is a linear combination of the columns of the matrix
=⎣⎡ADGBEHCFI⎦⎤⎣⎡xyz⎦⎤x⎣⎡ADG⎦⎤+y⎣⎡BEH⎦⎤+z⎣⎡CFI⎦⎤(2.7)