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Examples for
Step-by-Step Linear Algebra
Complex Numbers
See complex number addition with steps:
(9-8i)+(1+4i)
(1-i) plus (2i-3)
Perform complex number subtraction:
(2+3i)-(5-i)
Show the multiplication of complex numbers step by step:
(-2i+3) times (5i+4)
Find the magnitude of a complex number:
magnitude of 4+7i
norm of 2i+3
|4 - 6i|
Show the steps to rationalize a complex number:
rationalize -2/(7-3i)
rationalization of (-2i+1)/(i-1)
Vector Norms
Calculate a vector norm or length:
vector length {2,3,4}
norm of (3, 7, 9, 4)
See the steps to find the norm of a symbolic vector:
norm {a, 2b}
norm of (y, Sqrt(3), 3x)
Linear Independence
Determine the linear independence of vectors step by step:
linear independence {1,0,0},{2,0,0},{0,4,5}
are (1,9) and (2, 18) linearly independent?
Explain the linear independence of symbolic vectors:
when are (a, b) and (2a, 4b) linearly independent?
linear independence of [2, x, 6], [5y, 1, 0], [0, 0, 1]
Determinants
Find the determinant step by step with various methods:
determinant of {{1,2}, {-1, 2}}
determinant of {{1,2,1}, {1,1,0}, {0,1,1}}
Row Reduction
Write a matrix in reduced row echelon form one step at a time:
{{1,1,5},{1,-1,1}} row reduce
reduced row echelon form: {{1, -3, 3, -4}, {2, 3, -1, 15}, {4, -3, -1, 19}}
rref {{1.2, 5.6}, {3.2, 4.7}}
Perform row reduction on a complex matrix:
echelon form{{1-5i,4-i},{-1+2i,3+i}}
row reduce {{i, 2i+1, i+1}, {3-i, i-1, i+3}, {4i-2, 2i+1, 0}
Characteristic Polynomial
Find the characteristic polynomial of a matrix:
characteristic polynomial {{1, 2}, {-1, 4}}
char poly {{10,-35,50,-24},{1,0,0,0},{0,1,0,0},{0,0,1,0}}
Find the characteristic polynomial of a complex matrix step by step:
char polynomial {{i, 1, 2i-1}, {0, 2i+1, 2}, {3i-1, 0, i}}
Vector Arithmetic
See vector addition with steps:
(7, 11) + (-5, 9)
{4, 3, 8} + {2, 8, 1}
Perform vector subtraction:
(12, 53) - (19, 24)
[1, 5, 6] - [6, 5, 1]
Explain complex vector subtraction with steps:
(2i-1, i) minus (2i, i-1)
Multiply a vector by a scalar:
3*(1, 4, 5)
multiply {2,7,9} by -5
See how to compute a dot product:
(4, 1) . (-2, 3)
dot [1,2,9] [5,7,9]
Calculate the dot product for symbolic vectors step by step:
(a, b) dot (2a, 3b)
Compute a cross product:
(1,2,3)x(3,4,5)
cross product of {2, 4, 8} and {1, 3, 7}
See the steps to calculate the cross product of symbolic vectors:
cross {a, b, 2 c} {2 a, 3 b, c}
Distance between Vectors
See step by step how to find the distance between vectors:
distance between (6, 0, 2) and (2, 1, 4)
distance {3, 6} {1, 8}
dist (2, 4, 1) (-1,5, 0)
Calculate the distance between symbolic vectors:
distance {3b, 2a} {2b, a}
Matrix Arithmetic
See the steps to simplify the components of a matrix:
{{2, 2^2, 8/4}, {1, - 4, 7 - 4}}
{{2-1, 4*3}, {3^2, 8}}
simplify {{2b-2b,-7a+12a, a*a*a},{-a*a,3b-1b +1, 3}, {2*4b,8,b-b}}
{{2i-3i, (2+i)-(2i+1)}, {1-i-3, 2i+3-i}}
Show the steps to add matrices:
add {{1, -1}, {1, 1}} and {{1, -2}, {-3, 5}}
{{11,-6, 2},{-9,4,-8}} + {{1,8, 5}, {-2,-3, 16}}
Add complex matrices:
{{1-i, 2i+1}, {i, -2+2i}} + {{-i +3, 1-i}, {0, 3i-2}}
{{2i+3, 1, i-2}, {i^2, -2+i, 4i}, {i-1, 2i+3, 3i-1}} plus {{2i-1, 3i-2, 2i},{ i^2-i, i-2, 2i+3},{2-3i, i-2, 3-i}}
Explain the subtraction of matrices step by step:
{{3,2},{7,9}} subtract {{7,12}, {3,21}}
{{1, -3, 5}, {-7, 9, -11}} - {{-2, 4, -6}, {8, -10, 12}}
Show the subtraction of complex matrices:
{{2i-1, 3i+1}, {1-i, i-2}} minus {{1+i, 2i-2}, {3i, i-2}}
Perform multiplication of a matrix by a scalar:
7*{{2, 5}, {-7, 18}}
multiply {{2,7,-6}, {8,9,-14},{3,-1,4}} by -3
See the steps to multiply matrices:
{{1, 2}, {3, 4}} . {{-1, 1}, {0, 2}}
multiply {{4,8,0}, {3,-9,6}} and {{2,1}, {7,5}, {3,9}}
Show how to multiply symbolic matrices:
{{a, 2b}, {b-a, 3a}} . {{-b+a, b}, {2b, 2a}}
Explain complex matrix multiplication step by step:
{{2-i,1+3i},{3-2i,1}} times {{1-i,6+3i}, {2i-4, 2i}}
Minors
Find a specific minor of a matrix step by step:
value of the 2,1 minor of {{1,1,2},{3,5,8},{13,21,34}}
(1,2)-minor of {{1,2,3},{3,4,5},{5,4,3}}
See the steps to calculate a minor of a symbolic matrix:
M-23 of {{2,a,5,b},{7,c+d,9,11},{13,17,19,23},{a-1,-2,b+c,4}}
Show the steps to find a minor of a complex matrix:
compute the (2,2)-minor of {{I,1,I-2},{1,2I,1},{4,6,4-2I}}
Rank & Nullity
Find the rank of a matrix:
rank {{1, 2, 1}, {-2, -3, 1}, {3, 5, 0}}
Calculate the rank of a complex matrix:
rank {{1 + i, 2, 3 - 2i}, {0, 4, 5i}, {1 + i, 6, 3 + 3i}}
Find the nullity of a matrix:
nullity {{1, 2}, {3, 4}, {5, 6}, {7, 8}}
Calculate the nullity of a complex matrix:
nullity of {{i-1, 2i+1, i}, {2i-1, i+1, 3i-1}, {i+2, 2i-2, i+3}}
Eigenvalues & Eigenvectors
Compute eigenvalues and eigenvectors step by step:
eigenvalues {{3,-1},{0,2}}
eigenvectors {{7,0,-3},{-9,-2,3},{18,0,-8}}
eigensystem {{2,2,-3}, {2,1,-6}, {-1,-2,0}}
See the steps to compute the eigenvalues and eigenvectors of a complex matrix:
eigenvalues {{i-2, 2i}, {2i-1, i-1}}
eigenvectors {{1-i, 1, 2i}, {0, i-2, 1}, {2i-1, 0, 1}}
GO FURTHER
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Linear Algebra
Practice Problems
RELATED EXAMPLES
Linear Algebra
Matrices
Vectors
Angle between Vectors
Show the steps to find the angle between two vectors:
angle between {-1,3} and {4,9}
{-1, -3, -7} angle {4, 5, 6}
Calculate the angle between two symbolic vectors:
angle between [2a, b] and [a, 2b]
Trace
Find the trace of a matrix step by step:
trace of {{7,8,9}, {4,5,6}, {1,2,3}}
tr {{6,2,-3}, {-8,4,6}, {3,7,-11}, {7,4,-2}}
Calculate the trace of a symbolic matrix:
trace {{3b, 3a}, {b, 2a+2b}}
See the steps to find the trace of a complex matrix:
trace of {{i, 3i-2, 2i-2}, {1-2i, 4i-1, 2i}, {2i+1, 2, 2-3i}}
Inverse
Invert a matrix one step at a time:
inverse of {{1, 1, 2}, {-1, 2, 2}, {3, 2, 3}}
invert {{2,4}, {1,3}}
Show the steps to invert a symbolic matrix:
inverse {{a,b}, {2a,3b}}
Calculate the inverse of a complex matrix:
inverse of {{i+1, 2i+1, i-1}, {i+1, 2i-1, i-3}, {i-1, 2i, i-2}}
Null Space
Find the null space of a matrix:
null space {{1, 3, 3}, {-3, -5, -3}, {3, 3, 0}}
kernel {{0, 1, 0}, {-1, 0, 2}, {0, -1, 0}, {0, 0, -1}}
Show the steps to calculate the null space of a complex matrix:
null space of {{1 + i, 1 - i}, {-1 + i, 1 + i}}
Systems of Linear Equations
Solve linear systems using elimination, substitution, Gaussian elimination and Cramer's rule:
2x + y = -1, x – 4y = 3
x + 2y - z + w = 6, -x + y + 2z – w = 3, 2x – y + 2z + 2w = 14, x + y – z + 2w = 8
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