Multiply vectors in a matrix by a matrix

You can actually use Dot directly. It works in unintuitive ways with tensors.

n = 1000;
mat1 = Table[Table[RandomReal[1], {2}], {n}, {n}];

matRot = Map[RotationMatrix[3 Degree].# &,  mat1, {2}]; // AbsoluteTiming

{79.8211, Null}

matRot2 = Map[RotationTransform[3. Degree], mat1, {2}]; // AbsoluteTiming

{9.04459, Null}

matRot3 = mat1.RotationMatrix[3 Degree]\[Transpose]; // AbsoluteTiming

{0.072949, Null}

matRot == matRot2 == matRot3

True

EDIT: While this is not what you asked, might I suggest you build mat1 directly with RandomReal? Bottlenecks are probably elsewhere, but anyways.

SeedRandom[16];
mat1 = Table[Table[RandomReal[1], {2}], {n}, {n}]; // AbsoluteTiming

{0.638727, Null}

SeedRandom[16];
mat = RandomReal[1, {n, n, 2}]; // AbsoluteTiming

{0.015406, Null}

mat == mat1

True

n = 1000;
mat1 = Table[Table[RandomReal[1], {2}], {n}, {n}];

matRot = Map[RotationMatrix[3 Degree].# &,  mat1, {2}]; // AbsoluteTiming

{110.570004, Null}

matRot2 = Map[RotationTransform[3. Degree], mat1, {2}]; // AbsoluteTiming

{13.946951, Null}

matRot == matRot2

True

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