Basis

The bases for a linear space describe the space. Each member basis is independent.

Description

Bases are independent vectors that can be linearly combined to span an entire subspace. If a basis vector is removed, the subspace necessarily shrinks.

If the basis vectors are orthonormalized, they form an orthonormal basis. A convenient pair of orthonormal basis vectors in R² are [1 0] and [0 1]. A convenient set of orthonormal basis vectors in R³ are [1 0 0], [0 1 0], and [0 0 1]. And so on.

A null space has no basis.

All non-null spaces have infinitely many possible bases to choose from, because the only requirement on a basis is that it be independent.

Coordinatization

Given a space (like Rⁿ) and bases that span that space (the set B = u₁ ... u_n), any vector v in that space can be represented as a linear combination of the bases. In other words, there must be a set of coefficients c₁ ... c_n that satisfy:

c₁u₁ + ... + c_nu_n = v

The vector of coefficients (v_B = [c₁, ... c_n]) is called the coordinate vector relative to B.

By rewriting the set of bases as a matrix M_B, the above statement becomes:

M_B v_B = v

v_B can then be identified through elimination of the augmented matrix [u₁ ... u_n | v], or through most software packages as M_B \ v.

Change of Basis

A space can be linearly transformed to bring about a change of basis. This transformation can be expressed with matrix multiplication.

Following from the above notation, since the following are known to be true:

M_B' v_B' = v

M_B v_B = v

It must also be true that:

M_B' v_B' = M_B v_B

v_B' = M_B'^-1 M_B v_B

The change of basis matrix C_B,B' (note the subscript, indicating that it transforms from B to B') is defined as:

C_B,B' = M_B'^-1 M_B

This can then be identified through elimination of the augmented matrix [M_B' | M_B], or through most software packages as M_B' \ M_B.

Such a change of basis has a linear scaling effect on space. The scaling factor is the determinant. If the determinant is 0, then the matrix expresses a transformation that removes one (or more) basis vector(s). Such a transformation effectively collapses the space to a lower dimension.

Any matrix that has basis is invertible, and therefore has a non-zero determinant.

Usage

If a matrix is diagonalizable, identifying the change of basis that transforms it into a diagonal matrix enables several efficient strategies for solving systems.

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