Composition of Linear transformation & Matri Multiplication

3 important questions on Composition of Linear transformation & Matri Multiplication

What are some properties of the composition of linear transformations?

Let T be a linear transformation, and U1, U2 be in Left vector transformation. Then
-T(U1+U2)= TU1 + TU2
-(U1+U2)T= U1T + U2T
-T(U1U2)=(TU1)U2
-TI=IT=T
-a(U1U2)=(aU1)U2=U1(aU2)

What are properties of multiplication of linear transformations?

If V,W,Z are finite dimensional vector spaces with bases alpha, beta, gamma.
And T: V->W, U:W->Z are linear transformations
Then the matrix representation of UT with respect to alpha and gamma is [U]beta, gamma [T]alpha, beta

What is associativity of matrix multiplication?

If A,B,C are matrices such that  A(BC) is defined, then (AB)C is also defined and they are equal. A(BC)=(AB)C

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