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THALG 2Vector spaces

If $B=\{u_1,\ldots,u_n\}$ is a basis for $V$ , then every vector $v\in V$ has a unique representation

v=\lambda_1u_1+\cdots+\lambda_nu_n, \qquad \lambda_1,\ldots,\lambda_n\in F.

1Coordinates and linear structure / why coordinates are unique

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Coordinate vector

DFALG 2Vector spaces

Let $B=(u_1,\ldots,u_n)$ be an ordered basis for a vector space $V$ . If

u=x_1u_1+\cdots+x_nu_n,

then the coordinate vector of $u$ with respect to $B$ is the column

[u]_B= \begin{pmatrix} x_1\\ \vdots\\ x_n \end{pmatrix}.

1Coordinates and linear structure / coordinates in a basis

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Polynomial interpolation

DFALG 2Vector spaces

Given $n$ distinct inputs $t_1,\ldots,t_n$ and prescribed outputs $y_1,\ldots,y_n$ , polynomial interpolation finds the polynomial $f\in P_{n-1}$ satisfying

f(t_i)=y_i,\qquad i=1,\ldots,n.

Using the standard basis

B=(1,t,t^2,\ldots,t^{n-1}),

write

f(t)=a_0+a_1t+\cdots+a_{n-1}t^{n-1}.

The coefficients form the coordinate vector

[f]_B= \begin{pmatrix} a_0\\a_1\\\vdots\\a_{n-1} \end{pmatrix}.

1Coordinates and linear structure / coordinates of a polynomial
4Local approximation / match values instead of derivatives

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Matrix representation of a linear map

THALG 3Linear transformations

Let $T:V\to W$ be a linear map, where $\dim(V)=n$ and $\dim(W)=m$ . For ordered bases

B_V=(v_1,\ldots,v_n),\qquad B_W=(w_1,\ldots,w_m),

form the coordinate columns

a_j=[T(v_j)]_{B_W}\qquad (j=1,\ldots,n).

Then

A=\begin{pmatrix}a_1&\cdots&a_n\end{pmatrix}

is the unique $m\times n$ matrix satisfying

[T(v)]_{B_W}=A[v]_{B_V}\qquad\text{for every }v\in V.

We call $A$ the matrix representation of $T$ relative to $B_V$ and $B_W$ and may write it as $[T]_{B_V}^{B_W}$ .

With this notation, the subscript names the domain basis and the superscript names the codomain basis, matching the direction $[v]_{B_V}\mapsto[T(v)]_{B_W}$ in the displayed coordinate identity.

1Coordinates and linear structure / a linear map in coordinates

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