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A basis for a vector space is a sequence of vectors that form a set that is linearly independent and that spans the space.

A set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called basis vectors.

For example

The space of finite-degree polynomials has a basis with infinitely many elements.

$\left\{1,x,x^2,....\right\}$

To obtain the change of basis matrix ${\mathrm{S}}_{\mathrm{u}}鈫�\mathrm{B}$from u to B.

Since u = {f,g}, B = {f , f + g }

So,

f = 1 - f + 0 - (f + g) ....(1)

g = -1 - f + 1 - (f+g) ...(2)

Since the first vector of the base is f, then the first column of the change of basis matrix will be formed of the coordinates in (1) and the second column will be formed out of the coordinates (2).

So, $S=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$.

By the above statement, this matrix is not the one given in the text.