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:
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: