## Linear Algebra: A Modern Introduction

$$5a$$
We're given the expression $$2(a-3b) + 3(2b+a)$$ By the distributive property of scalar multiplication over vector addition (Theorem 1.1.e), we get: $$2a+2(-3)b + 3(2b)+3a$$ By the associative property of scalar-vector multiplication (Theorem 1.1.g), we get: $$2a-6b + 6b+3a$$ Using the commutativity (Theorem 1.1.a) and associativity (Theorem 1.1.b) of vector addition, we rearrange to get: $$(2a+3a) + (-6b + 6b)$$ By applying the distributive property of vector-scalar multiplication over scalar addition (Theorem 1.1.f), we get: $$(2+3)a + (-6+6)b$$ With scalar addition, we condense to: $$5a + 0b$$ And by the identity property of vector addition (Theorem 1.1.c), we get $$5a$$