Maple examples for linear algebra
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# signs introduce comments (you don't need to type these)

M := Matrix([[1, 2, 3], [4, 5, 6]]);  # by rows

A := <<0, -2, -2, -2> | <-3, 1, 1, -3> | <1, -1, -1, 1> | <2, 2, 2, 4>>; # by columns

M := Matrix(3, 3, [[4, 7, 3], [3, 0, 6], [2, 2, 4]]);
M.M; # matrix multiplication

N := Matrix(3, 5, [[4, 7, 3, 3, -22], [3, 0, 6, 7, 8], [2, 2, 4, 11, 6]]);
ReducedRowEchelonForm(N);
HermiteForm(N);

HermiteForm(N, output=['H', 'U']) # get transformation matrix as well (H = U.N)

N2 := Matrix([N, IdentityMatrix(3)]); # pad N on the right
E2 := ReducedRowEchelonForm(E2);
E := SubMatrix(E2, 1 .. 3, 1 .. 5);
U := SubMatrix(E2, 1 .. 3, 6 .. 8); # the transformation matrix
Equal(U . N, E); # should return "true"


A := <<0, -2, -2, -2> | <-3, 1, 1, -3> | <1, -1, -1, 1> | <2, 2, 2, 4>>;
JordanForm(A, output = ['J', 'Q']);

J,Q = JordanForm(A, output = ['J', 'Q']); # store the answer
Equal(A, Q.J.Q^(-1)); # should return "true"

B := <<0, 32, -2, -2> | <-3, 1, 1, -3> | <1, -1, -1, 1> | <2, 2, 2, 4>>











Sage examples for linear algebra
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# again, signs introduce comments (you don't need to type these)
#Vectors and matrices
A = Matrix(QQ, [[1,2,3],[3,2,1],[1,1,1]])   
A[0,0]
A.det()
A.echelon_form()
matrix.identity(3)
B=A.augment(_). #Pad _ on the left
C=B.rref()
B[[0,1,2],[3,4,5]] #Extract the transformation matrix  

matrix(QQ,2,[1,2,3,4,5,6]); #Input matrix by only giving one vector

A.left_kernel()
A.right_kernel()
w=vector(QQ, [1,2,4])
A*w
w*A
##The matrix A was  created as a matrix over the rationals. If no ring is specified, sage tries to choose an appropriate ring. For example
v=vector([1,2,4])
# is created as a vector over the integers, and 
x=vector([1,2,1/4])
x.base_ring()
# is created as a vector over the rationals. 
A*v
x*A
### Now let us consider A over F_2
 B=A.change_ring(GF(2)) 
 A.base_extend(GF(2))
B.base_extend(GF(4))
B*x 
B*v

### Now let us consider A over Z
C=A.change_ring(ZZ)
C.smith_form() 
C.hermite_form()  

###Jordan form
A.jordan_form()  
## But what if it does not exist over Q? You can go to Qbar
A=Matrix([[0,-1],[1,0]]
A.change_ring(QQbar)   
_.jordan_form()


###########Vectorspaces
V = VectorSpace(QQ,4)
S = V.subspace([V([1,1,0,0]),V([1,0,0,0]), V([0,1,0,0])])

T = V.subspace([V([1,2,3,4]),V([5,6,7,8]), V([9,1,0,1])])
T.intersection(S)      
T+S 
_==V


W = span([[1,2,3], [4,5,6]], ZZ)