What does it mean when a matrix is equal to zero?

When the determinant of a matrix is zero, the volume of the region with sides given by its columns or rows is zero, which means the matrix considered as a transformation takes the basis vectors into vectors that are linearly dependent and define 0 volume.

What does it mean when the dot inner product of two vectors is equal to zero?

orthogonal
Equivalently, it is the product of their magnitudes, times the cosine of the angle between them. The dot product of a vector with the zero vector is zero. Two nonzero vectors are perpendicular, or orthogonal, if and only if their dot product is equal to zero.

What happens when you multiply a matrix by the zero vector?

Well, any zero matrix multiplied to a vector will have as a result a zero vector. That is, if the dimensions of the matrix and the vector follow the rules of matrix multiplication, in other words, if the multiplication can be defined, then the result will certainly be a zero vector.

Does the zero matrix have a null space?

Basically you need to get A into row reduced echelon form. You have 2 equations, 6 unknowns (so you have 4 free parameters.) where a,b,c,d are your column vectors which give the coefficents of your x3,x4,x5,x6 when you solve for these variables.

What does it mean when inner product is 0?

Definition: Two vectors are orthogonal to each other if their inner product is zero. That means that the projection of one vector onto the other “collapses” to a point. So the distances from to or from to should be identical if they are orthogonal (perpendicular) to each other.

What does the inner product represent?

An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.

What happens when a matrix has a row of zeros?

A matrix is in reduced row-echelon form when all of the conditions of row-echelon form are met and all elements above, as well as below, the leading ones are zero. If there is a row of all zeros, then it is at the bottom of the matrix. The first non-zero element of any row is a one.

What do you multiply a matrix by to get a zero matrix?

Just as any number multiplied by zero is zero, there is a zero matrix such that any matrix multiplied by it results in that zero matrix.

When is the cross product of two vectors zero?

The cross product of two vectors are zero vectors if both the vectors are parallel or opposite to each other. Conversely, if two vectors are parallel or opposite to each other, then their product is a zero vector. Two vectors have the same sense of direction.

What is the difference between a vector and a matrix?

Vectors and Matrices. A vector, u, means a list (or n-tuple) of numbers: u = (u 1, u 2, . . . , u n) where u i are called the components of u. If all the u i are zero i.e., u i = 0, then u is called the zero vector. Given vectors u and v are equal i.e., u = v, if they have the same number of components and if corresponding components are equal.

What is a cross product in math?

What is a Cross Product? Cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both vectors. The Vector product of two vectors, a and b, is denoted by a × b.

What is the vector product of two vectors called?

Cross Product of Two Vectors The vector product or cross product of two vectors A and B is denoted by A × B, and its resultant vector is perpendicular to the vectors A and B.