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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)

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}}

See the steps to calculate the minors matrix:

matrix of minors of {{4,5,6,1},{0,5,6,1},{0,0,6,1}, {0,0,0,1}}
minors of {{a,0,1}, {3, c, 3}, {1,8,9}}

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}}

Matrix Decompositions

See the steps to find the LU decomposition of a matrix:

LU decomposition {{1,1}, {0,2}, {3,3}}
lu decomposition ((2,3,3), (6,7,9))

Explore how to compute the Cholesky decomposition on positive definite Hermitian matrices:

what is the Cholesky decomposition of {{8,1},{1,4}}
Cholesky Decomposition ({{1, I}, {-I, 2}})

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}

Vector Normalization

See the steps to normalize a vector:

normalize the vector (0,5,3,2)
normalize 3i-2j+k

Normalize a symbolic vector step by step:

(6,5,a) normalized
normalize the vector {3a,-2b}

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}}

Cofactors

See the steps to find the cofactor matrix of a matrix:

cofactors of {{1,2,3}, {2,3,4}, {3,4,5}}
cofactor matrix of {{1,2},{0,3}}
matrix of cofactors {{1,2,0}, {2,7,-1}, {1,5,7}}

Find a specific cofactor of a matrix step by step:

cofactor (2,1) of {{1,1,2},{3,5,8},{13,21,34}}
compute the cofactor (1,1) of {{1,2,3},{3,4,5},{5,4,3}}
value of the cofactor (1,2) of {{0,-1}, {-1,0}}

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

Free Unlimited Linear Algebra Practice Problems

RELATED EXAMPLES

  • Linear Algebra
  • Matrices
  • Vectors

    • Orthonormalization

    See the steps to orthonormalize a set of vectors:

    Orthogonalize {-1, 2, I}, {2, 3, 4}, {5, 6, 7}
    Gram-Schmidt process for {1,2},{5,6},{9,9},{8,9}

    Find an orthonormal basis for a set of complex vectors:

    Orthonormal basis of {{-1, 2, I}, {2, 3, 2}, {1, 2I, 1}}
    Orthonormalize {{2, I}, {0, I}}

    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}}

    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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