# Linear Algebra - Questions with Solutions

Linear algebra questions with solutions and detailed explanations on matrices , spaces, subspaces and vectors , determinants , systems of linear equations
and online linear algebra calculators are included.

## Matrices

Matrices with Examples and Questions with Solutions .Transpose of a Matrix .

Symmetric Matrix .

Identity Matrix .

Diagonal Matrices .

Triangular Matrix .

Diagonalization of Matrices .

Inverse Matrix Questions with Solutions .

Find Matrix Inverse Using Row Operations

Add, Subtract and Scalar Multiply Matrices .

Multiplication and Power of Matrices

Eigenvalues and Eigenvectors Questions with Solutions

Row Operations and Elementary Matrices .

The Three Row Operations on Augmented Matrices .

Write a Matrix in Reduced Row Echelon Form .

Pivots of a Matrix in Row Echelon Form - Examples with Solutions .

Null Space and Nullity of a Matrix .

Column and Row Spaces and Rank of a Matrix .

Free and Basic Variables of a Matrix - Examples with Solutions .

Orthogonal Matrices - Examples with Solutions .

The QR Decomposition of a Matrix .

LU Decomposition of a Matrix .

Properties of Matrix Operations .

## Least Squares Problems

Solve Least Squares Problems by the Normal Equations .Solve Least Squares Problems by the QR Decomposition .

## Spaces, Subspaces and Vectors

Vector Spaces - Examples with Solutions .Subspaces - Examples with Solutions .

Vectors in ℝ

^{n}.

Inner Product, Orthogonality and Length of Vectors .

Orthogonal Vectors - Examples with Solutions .

Linear Combinations and Span of Vectors .

Linearly Independent and Dependent Vectors - Examples with Solutions .

Testing for Linearity of Vectors in a Subspace - Examples with Solutions .

Basis, Coordinates and Dimension of Vector Spaces .

Change of Basis - Examples with Solutions .

Orthonormal Basis - Examples with Solutions .

The Gram Schmidt Process for Orthonormal Basis . Examples with Solutions

## determinants

Determinant of a Square Matrix .Find Determinant Using Row Reduction .

## Systems of Linear Equations

The Three Elementary Operations on Systems .Gaussian Elimination to Solve Systems - Questions with Solutions .

The Elimination Method in Systems - Questions with Solutions .

Cramer's Rule with Questions and Solutions .

## Linear Algebra Calculators

Find the Inverse of a Matrix Using Row Reduction .Row Echelon Form Calculator .

Row Reduce Agmented Matrices - Calculator .

Online Calculator for The Determinant of a Matrix of Any Size .

QR Decomposition of Matrices Calculator .

Online LU Decomposition of a Matrix Calculator .

Systems of Equations with Complex Coefficients Solver .