Pro

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:

Perform row reduction on a complex matrix:

Characteristic Polynomial

Find the characteristic polynomial of a matrix:

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:

Compute a cross product:

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:

Add complex matrices:

Explain the subtraction of matrices step by step:

Show the subtraction of complex matrices:

Perform multiplication of a matrix by a scalar:

See the steps to multiply matrices:

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:

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:

GO FURTHER

Free Unlimited Linear Algebra Practice Problems

RELATED EXAMPLES

  • Linear Algebra
  • Matrices
  • Vectors
  • 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:

    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:

    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:

    Go Pro NowLearn more about Wolfram|Alpha Pro »