Linear Equations
The condition that guarantees a unique solution for a system of linear equations is that the coefficient matrix has this property.
What is full rank?
The identity matrix is the only matrix that commutes with all matrices of the same size, except for these.
What are scalar multiples of the identity?
Two nonzero vectors in Rn are orthogonal if and only if this condition holds.
What is their dot product equals zero?
A set of vectors is linearly dependent if and only if one vector can be written as this.
What is a linear combination of the others?
A linear transformation from Rn to Rm can always be represented by this object
What is an m×n matrix?
A system with no solution is called this.
What is an inconsistent system?
The inverse of a product of two invertible matrices A and B is given by this expression.
What is B^(−1)A^(−1)?
The magnitude of the cross product of two vectors equals the area of this geometric figure.
What is a parallelogram?
The span of a set of vectors is defined as this.
What is the set of all linear combinations of those vectors?
The kernel of a linear transformation is a subspace of this space.
What is the domain?
A system is consistent if and only if the rank of the coefficient matrix equals this rank.
What is the rank of the augmented matrix?
A matrix is invertible if and only if its determinant satisfies this condition.
What is being nonzero?
The scalar triple product of three vectors equals the signed volume of this shape.
What is a parallelepiped?
A basis for a vector space must satisfy these two properties.
What is spanning and linear independence?
A linear transformation is invertible if and only if this condition holds for its matrix representation
What is having full rank (or nonzero determinant if square)?
If a system has more variables than equations, the solution set is typically described as this geometric object.
What is a subspace?
The determinant of a triangular matrix equals this.
What is the product of its diagonal entries?
The number of vectors in any basis for a vector space is called this.
What is the dimension?
Eigenvectors corresponding to distinct eigenvalues of a linear operator on Rn have this property.
What is linear independence?
The determinant must be equal to this for a square system to have a unique solution.
What is not zero?
The determinant of a matrix can be computed using these two methods: row operations and this expansion technique.
What is cofactor expansion?
The determinant of the matrix whose columns are three vectors in R3 equals this geometric quantity, up to sign.
What is the volume of the parallelepiped formed by the vectors?
This theorem states that for any linear map, the sum of the rank and nullity equals the dimension of the domain.
What is the rank-nullity theorem?
The matrix representation of a linear transformation changes under this operation.
What is a change of basis?