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MATH
[1251]

Theorems from Geometric examples

All theorems from Geometric examples.

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Matrix maps are linear

For every m×nm\times n matrix AA, the map

TA:Rn→Rm,TA(x)=AxT_A:\mathbb R^n\to\mathbb R^m,\qquad T_A(x)=Ax

is linear.

  1. [1]:Linear map
  2. [2]:Proof
  3. [3]:Reading a matrix map in coordinates
  4. [4]:The matrix of a linear map on standard coordinates
  5. [5]:Scaling in R2\mathbb R^2
  6. 4 more

The matrix of a linear map on standard coordinates

Let T:Rn→RmT:\mathbb R^n\to\mathbb R^m be a linear map and let e1,…,ene_1,\ldots,e_n be the standard basis vectors of Rn\mathbb R^n. If AA is the m×nm\times n matrix whose jjth column is T(ej)T(e_j), then

T(x)=AxT(x)=Ax

for every x∈Rnx\in\mathbb R^n. Thus every linear map from Rn\mathbb R^n to Rm\mathbb R^m is represented by a matrix.

  1. [1]:Linear map
  2. [2]:Preservation of finite linear combinations
  3. [3]:Matrix maps are linear
  4. [4]:Proof
  5. [5]:Constructing the matrix from a formula

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Matrix maps are linearThe matrix of a linear map on standard coordinates