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MATH
[1251]

Theorems from Matrix representations

All theorems from Matrix representations.

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Matrix representation of a linear map

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

BV=(v1,…,vn),BW=(w1,…,wm),B_V=(v_1,\ldots,v_n),\qquad B_W=(w_1,\ldots,w_m),

form the coordinate columns

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

Then

A=(a1⋯an)A=\begin{pmatrix}a_1&\cdots&a_n\end{pmatrix}

is the unique m×nm\times n matrix satisfying

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

We call AA the matrix representation of TT relative to BVB_V and BWB_W and may write it as [T]BVBW[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]BV↦[T(v)]BW[v]_{B_V}\mapsto[T(v)]_{B_W} in the displayed coordinate identity.

The jjth column is the coordinate vector of T(vj)T(v_j) in the codomain basis. Thus the number of columns is the dimension of the domain and the number of entries in each column is the dimension of the codomain.

  1. [1]:Linear map
  2. [2]:Basis
  3. [3]:Coordinate vector
  4. [4]:Values on a basis determine a linear map
  5. [5]:Diagonalisation
  6. [6]:Diagonalisation
  7. [7]:Characteristic polynomial
  8. [8]:Eigenvalues and eigenvectors
  9. 10 more

Equivalent invertibility conditions

For a linear map T:V→WT:V\to W between finite-dimensional spaces, the following conditions are equivalent when dim⁡(V)=dim⁡(W)\dim(V)=\dim(W):

  1. TT is invertible;
  2. TT is bijective;
  3. TT has a left inverse;
  4. TT has a right inverse.

A left inverse makes TT injective and a right inverse makes TT surjective. In equal dimensions either property makes TT bijective, so the one-sided inverse is then the two-sided inverse of TT.

  1. [1]:Bijective map
  2. [2]:Inverse map
  3. [3]:Bijectivity in equal dimensions
  4. [4]:Inverse maps in coordinates

Inverse maps in coordinates

Let VV and WW be finite-dimensional vector spaces, let T:V→WT:V\to W be bijective and fix ordered bases BVB_V of VV and BWB_W of WW. Then T−1:W→VT^{-1}:W\to V is linear. If

A=[T]BVBW,A=[T]_{B_V}^{B_W},

then AA is square and

[T−1]BWBV=A−1.[T^{-1}]_{B_W}^{B_V}=A^{-1}.

Equivalently, for every w∈Ww\in W,

[T−1(w)]BV=A−1[w]BW.[T^{-1}(w)]_{B_V}=A^{-1}[w]_{B_W}.

Conversely, if the matrix representation AA is invertible, then TT is bijective.

The formula follows by writing w=T(v)w=T(v), so that [w]BW=A[v]BV[w]_{B_W}=A[v]_{B_V} and hence [v]BV=A−1[w]BW[v]_{B_V}=A^{-1}[w]_{B_W}. The two compositions are represented by A−1AA^{-1}A and AA−1AA^{-1}, which return the original coordinate vector. To check that T−1T^{-1} is linear, write w1=T(v1)w_1=T(v_1) and w2=T(v2)w_2=T(v_2). Then T(αv1+βv2)=αw1+βw2T(\alpha v_1+\beta v_2)=\alpha w_1+\beta w_2 and uniqueness of preimages gives T−1(αw1+βw2)=αT−1(w1)+βT−1(w2)T^{-1}(\alpha w_1+\beta w_2)=\alpha T^{-1}(w_1)+\beta T^{-1}(w_2).

  1. [1]:Matrix representation of a linear map
  2. [2]:Properties read from a matrix representation
  3. [3]:Bijectivity in equal dimensions
  4. [4]:Equivalent invertibility conditions
  5. [5]:Inverse map
  6. [6]:A scalar map and its inverse

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Matrix representation of a linear mapEquivalent invertibility conditionsInverse maps in coordinates