Gram-Schmidt Process and Vector Spaces

To find the inverse of a matrix

To transform a set of vectors into an orthogonal set

To calculate the determinant of a matrix

To solve linear equations

Add all vectors together

Find the inverse of the first vector

Set the first new vector equal to the first basis vector

Calculate the determinant

The sum of all basis vectors

The sum of inner products to subtract non-orthogonal components

The sum of all vector lengths

The sum of all orthogonal vectors

4

1

2

3

By dividing each vector by its length

By subtracting a constant from each vector

By multiplying each vector by its length

By adding a constant to each vector

√2

2

√3

1

(1, 0, 1)

(1/3, 2/3, 1/3)

(-√2/2, 0, √2/2)

(-1/2, 0, 1/2)

It increases the number of vectors

It reduces the number of vectors

It simplifies calculations in vector spaces

It makes vectors dependent

It only works for 2D vectors

It requires vectors to be orthogonal initially

It can be applied to any set of basis vectors

It only works for complex numbers

The dimension of the space

The type of vectors used

The method of computing the inner product

The number of vectors