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Examples for

# Step-by-Step Linear Algebra

Complex Numbers#### See complex number addition with steps:

#### Perform complex number subtraction:

#### Show the multiplication of complex numbers step by step:

#### Find the magnitude of a complex number:

#### Show the steps to rationalize a complex number:

Vector Norms#### Calculate a vector norm or length:

#### See the steps to find the norm of a symbolic vector:

Linear Independence#### Determine the linear independence of vectors step by step:

#### Explain the linear independence of symbolic vectors:

Determinants#### Find the determinant step by step with various methods:

Row Reduction#### Write a matrix in reduced row echelon form one step at a time:

reduced row echelon form: {{1, -3, 3, -4}, {2, 3, -1, 15}, {4, -3, -1, 19}} #### Perform row reduction on a complex matrix:

Characteristic Polynomial#### Find the characteristic polynomial of a matrix:

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:

Vector Arithmetic#### See vector addition with steps:

#### Perform vector subtraction:

#### Explain complex vector subtraction with steps:

#### Multiply a vector by a scalar:

#### See how to compute a dot product:

#### Calculate the dot product for symbolic vectors step by step:

#### See the steps to calculate the cross product of symbolic vectors:

Distance between Vectors#### See step by step how to find the distance between vectors:

#### Calculate the distance between symbolic vectors:

Matrix Arithmetic#### See the steps to simplify the components of a matrix:

#### Show the steps to add matrices:

{{11,-6, 2},{-9,4,-8}} + {{1,8, 5}, {-2,-3, 16}} #### Add complex matrices:

{{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:

{{1, -3, 5}, {-7, 9, -11}} - {{-2, 4, -6}, {8, -10, 12}} #### Show the subtraction of complex matrices:

#### Perform multiplication of a matrix by a scalar:

#### See the steps to multiply matrices:

multiply {{4,8,0}, {3,-9,6}} and {{2,1}, {7,5}, {3,9}} #### Show how to multiply symbolic matrices:

#### Explain complex matrix multiplication step by step:

Minors#### Find a specific minor of a matrix step by step:

#### 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:

Rank & Nullity#### Find the rank of a matrix:

#### Calculate the rank of a complex matrix:

#### Find the nullity of a matrix:

#### Calculate the nullity of a complex matrix:

Eigenvalues & Eigenvectors#### Compute eigenvalues and eigenvectors step by step:

#### See the steps to compute the eigenvalues and eigenvectors of a complex matrix:

Angle between Vectors#### Show the steps to find the angle between two vectors:

#### Calculate the angle between two symbolic vectors:

Trace#### Find the trace of a matrix step by step:

tr {{6,2,-3}, {-8,4,6}, {3,7,-11}, {7,4,-2}} #### Calculate the trace of a symbolic matrix:

#### See the steps to find the trace of a complex matrix:

Inverse#### Invert a matrix one step at a time:

#### Show the steps to invert a symbolic matrix:

#### Calculate the inverse of a complex matrix:

Null Space#### Find the null space of a matrix:

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:

Systems of Linear Equations#### Solve linear systems using elimination, substitution, Gaussian elimination and Cramer's rule: