For first year students Stroud is probably a better maths textbook to work from, it walks you through everything more slowly.

L&L is a classic, and should be within reach for someone who has completed Kreyzigs book.

Renowned professor and author Gilbert Strang demonstrates that linear algebra is a fascinating subject by showing both its beauty and value. While the mathematics is there, the effort is not all concentrated on proofs. Strang's emphasis is on understanding. He explains concepts, rather than deduces. This book is written in an informal and personal style and teaches real mathematics. The gears change in Chapter 2 as students reach the introduction of vector spaces. Throughout the book, the theory is motivated and reinforced by genuine applications, allowing pure mathematicians to teach applied mathematics.

For a maths book that's a bit more targeted towards Chemical Engineers Bird Stewart is generally accepted as a good book.

Targeted at engineers, this book works well as an accompanying mathe book

Kreyzigs book does a good job of holding your hand through all the exercises.

A good book to cover vector analysis the modern way.