- #1

DumpmeAdrenaline

- 78

- 2

The book I am studying from presents vector subspace as an infinite collection of vectors in a vector space with the properties of additive and multiplicative closure and basis vectors as a way to characterize/write a subspace compactly. All the vectors in the subspace can be written as linear combinations of this subset of vectors. If this subset of vectors is a minimal set meaning it forms a subset equal to the vector space itself through appropriate linear combinations of the basis vectors) then this subset is called a basis vector. Lets add context to the above

Let X be an m*n data matrix, where m represents the number of samples and n represents the values of the variables of interest for the system we are studying (e.g a reactor). Suppose interested in determining the number of independent variables and independent samples.

Is the row space a subset of a subspace comprising of m row vectors? [x1,..xm] where x1=(a11,a12,..a1n) x2=(a21,a22,..a2n) and xm=(am1,am2,..amn).

If we perform a LU decomposition of matrix X

LPX=U

If there are r rows r<= m in U then the first rows form the basis vectors set for the row space. Specifically, all vectors within the subspace can be expressed as linear combinations of these r rows. Does this imply if we have the variables for a new sample we can represent it as a linear combination of the aforementioned basis row vectors (old samples)?

Let X be an m*n data matrix, where m represents the number of samples and n represents the values of the variables of interest for the system we are studying (e.g a reactor). Suppose interested in determining the number of independent variables and independent samples.

Is the row space a subset of a subspace comprising of m row vectors? [x1,..xm] where x1=(a11,a12,..a1n) x2=(a21,a22,..a2n) and xm=(am1,am2,..amn).

If we perform a LU decomposition of matrix X

LPX=U

If there are r rows r<= m in U then the first rows form the basis vectors set for the row space. Specifically, all vectors within the subspace can be expressed as linear combinations of these r rows. Does this imply if we have the variables for a new sample we can represent it as a linear combination of the aforementioned basis row vectors (old samples)?