Sage: 矩陣、線性方程組¶

Sage: Matrices and linear equations

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This work by Jephian Lin is licensed under a Creative Commons Attribution 4.0 International License.

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建構矩陣¶

Construct a matrix

  1. matrix( list of lists )：把 list of lists 中的每個 list 當作矩陣的列。
  2. matrix(r, list)：把 list 切成 r 個列。
  3. identity_matrix(n)：單位矩陣。
  4. zero_matrix(n) or zero_matrix(m,n)：全零矩陣。

利用 print（純文字）或 show（格式化文字）來顯示矩陣。

  1. matrix( list of lists ): return the matrix whose rows are the lists in list of lists .
  2. matrix(r, list): split list into r rows and return the matrix.
  3. identity_matrix(n): the identity matrix.
  4. zero_matrix(n) or zero_matrix(m,n): the zero matrix.

You may use print (pure text) or show (formatted output) to display the matrix.

In [ ]:
A = matrix([[1,2,3], 
            [4,5,6]])
print(A)
In [ ]:
A = matrix(2, [1,2,3,4,5,6])
show(A)
In [ ]:
A = identity_matrix(3)
# A = zero_matrix(3)
# A = zero_matrix(3,4)
show(A)

從矩陣中選取各項或子矩陣¶

Select an entry or a submatrix from a matrix

若 A 是一個矩陣。

  1. A[i,j]：選取第 ij 項。
  2. A[list1, list2]：選取列在 list1 中、行在 list2 中的子矩陣。

也可以混合使用﹐如 A[i, list] 或 A[list, j]。

Suppose A is a matrix.

  1. A[i,j]: return the ij-entry.
  2. A[list1, list2]: return the submatrix induced on rows in list1 and columns in list2 .

You may mix the two usages, such as A[i, list] or A[list, j] .

In [ ]:
A = matrix(2, [1,2,3,4,5,6])
show(A)
print(A[0,1])
show(A[[0,1],[1,2]])

選取子矩陣中 list 的可以用 a:b 的格式取代。

  1. a:b：從 a 到 b（不包含 b）。
  2. a:：從 a 到底。
  3. :b：從頭到 b（不包含 b）。
  4. :：全部。

The list used for selecting the submatrix can be replaced by a:b .

  1. a:b: from a to b (excluding b).
  2. a:: from a to the end.
  3. :b: from the beginning to b.
  4. :: all.
In [ ]:
A = matrix(2, [1,2,3,4,5,6])
show(A[:,1:])

可以把選出來的子矩陣設定成給定的矩陣。

You may assign new values to the selected submatrix.

In [ ]:
A = zero_matrix(2,3)
show(A)
A[0,0] = 100
show(A)
A[:,1:] = identity_matrix(2)
show(A)

求解¶

Finding the solution(s)

若 A 為一矩陣﹐可以用

  1. A.right_kernel() 或
  2. A.right_kernel().basis_matrix()

來找到零解的基底。

Let A be a matrix. You may use

  1. A.right_kernel() or
  2. A.right_kernel().basis_matrix()

to find a basis of the homogeneous solutions.

In [ ]:
A = matrix(2, [1]*6)
show(A)
print(A.right_kernel())
show(A.right_kernel().basis_matrix())

可以用 vector([list]) 來設定向量。
若 A 是矩陣而 b 是向量﹐
則 A \ b 會給出一組解。
（也可以用 A.solve_right(b)。）

Use vector([list]) to construct a vector. If A is a matrix and b is a vector, then A \ b returns a solution. (Equivalently, A.solve_right(b) has the same effect.)

In [ ]:
A = matrix(2, [1]*6)
b = vector([1,1])
show(A)
print(b)
print(A \ b)
print(A.solve_right(b))

最簡階梯形式矩陣、增廣矩陣¶

Reduced echelon form and augmenting matrix

若 A 為一矩陣﹐則 A.rref() 會回傳其最簡階梯形式矩陣。

Let A be a matrix. Then A.rref() returns its reduced echelon form.

In [ ]:
A = matrix(3, list(range(12)))
show(A)
show(A.rref())

若 A 為一矩陣而 b 為一向量﹐則 A.augment(b) 會回傳相對應的增廣矩陣。

Let A be a matrix and b a vector. Then A.augment(b) returns their augmenting matrix.

In [ ]:
A = matrix(3, list(range(12)))
b = vector([1,5,9])
Ab = A.augment(b)
show(Ab)
Ab = A.augment(b, subdivide=True)
show(Ab)

可以用以下函數對矩陣（或增廣矩陣）A 執行列運算。

  1. A.swap_rows(i,j) .
  2. A.rescale(i, k) .
  3. A.add_multiple_of_row(i, j, k) .

注意這些函數會直接修改矩陣本身而不會回傳任何矩陣。

Apply the row operation to a matrix (or an augmenting matrix) A by the following functions.

  1. A.swap_rows(i,j) .
  2. A.rescale(i, k) .
  3. A.add_multiple_of_row(i, j, k) .

Notice that these functions update the matrix itself and do not return any matrix.

In [ ]:
A = matrix(3, list(range(12)))
show(A)
print(A.swap_rows(0,1))
show(A)

反矩陣¶

Inverse matrix

若 A 和 B 都是矩陣﹐則 A.augment(B) 也可以建立其相對應的增廣矩陣。
藉此可以計算反矩陣。

Let A and B be matrices. Then A.augment(B) also returns the corresponding augmenting matrix. We may use it to compute the inverse.

In [ ]:
A = matrix([[1,1,1], 
            [1,2,4], 
            [1,3,9]])
I3 = identity_matrix(3)
AI = A.augment(I3, subdivide=True)
show(AI)
IB = AI.rref()
show(IB)
B = IB[:,3:]
show(A * B)

若 A 是方陣﹐

  1. A.is_invertible() 可以判斷其是否可逆﹐
  2. A.inverse() 會求出它的逆矩陣。

Suppose A is a square matrix. Then

  1. A.is_invertible() determines if A is invertible, and
  2. A.inverse() returns the inverse of A .
In [ ]:
A = matrix([[1,1,1], 
            [1,2,4], 
            [1,3,9]])
print(A.is_invertible())
show(A.inverse())

四大基礎子空間的標準基底¶

The standard bases of the four fundamental subspaces

lingeo 是為這份教材所寫的函式庫。
其內容包含在 lingeo.py 裡。
可以用

from lingeo import random_good_matrix

來匯入須要的函數。

lingeo is a library written for LA-notebook. Its content can be found in lingeo.py , and you may use the syntax like

from lingeo import random_good_matrix

to import the necessary functions.

In [ ]:
from lingeo import random_good_matrix
from lingeo import row_space_matrix, column_space_matrix, kernel_matrix, left_kernel_matrix

若 A 為一矩陣﹐則

  1. row_space_matrix(A) 的列是由 $\beta_R$ 組成、
  2. kernel_matrix(A) 的行是由 $\beta_K$ 組成、
  3. column_space_matrix(A) 的行是由 $\beta_C$ 組成、
  4. left_kernel_matrix(A) 的列是由 $\beta_L$ 組成。

Let A be a matrix. Then

  1. row_space_matrix(A) is the matrix whose rows are vectors in $\beta_R$,
  2. kernel_matrix(A) is the matrix whose columns are vectors in $\beta_K$,
  3. column_space_matrix(A) is the matrix whose columns are vectors in $\beta_C$, and
  4. left_kernel_matrix(A) is the matrix whose rows are vectors in $\beta_L$.
In [ ]:
m,n,r = 3,5,2
A = random_good_matrix(m,n,r)
show(A)

print("row space matrix:")
show(row_space_matrix(A))

print("kernel matrix:")
show(kernel_matrix(A))
In [ ]:
m,n,r = 3,5,2
A = random_good_matrix(m,n,r)
show(A)

print("column space matrix:")
show(column_space_matrix(A))

print("left kernel matrix:")
show(left_kernel_matrix(A))