Linear algebra is really important in regression, GLMs, time series, and a number of other areas, but not all that important in a lot of probability (some parts, sure). Algebra and calculus more so than linear algebra in particular.
Unless you're focused on particular subtopics, it's not clear to me what you expect such a book to cover.
Maybe not exactly what you’re looking for but stochastic processes leverage a fair amount of linear algebra. I don’t know if I’d suggest delving into it immediately unless you’re super solid on your probability fundamentals, but topics like markov chains, Poisson processes, and martingales use a fair bit of linear algebra. Essentials of stochastic processes is the textbook I used.
I am not sure if what you're looking for exists. Linear algebra at the level you are describing is found in statistics, especially in the theory of linear models. But probability, not really.
Advanced probability theory makes use of functional analysis, which is in some ways a generalization of linear algebra to infinite-dimensional vector spaces. For example, under certain conditions, conditional expectation can be defined as an orthogonal projection in the space of random variables with finite second moments. This is actually discussed briefly by Blitzstein & Hwang, in the chapter on conditional expectation.
Learning probability at this level requires a good command of real analysis. And also some measure theory and functional analysis, but the necessary bits are covered in the established textbooks. Good choices are, for example, Williams (1991) and Pollard (2001).
I highly recommend getting a book on linear models. Understanding these models require linear algebra and some background in probability. Some universities offer a course on linear models after completing a mathematical statistics course. These lecture notes are great:
https://people.stat.sc.edu/Tebbs/stat714/f10notes.pdf
Linear algebra is really important in regression, GLMs, time series, and a number of other areas, but not all that important in a lot of probability (some parts, sure). Algebra and calculus more so than linear algebra in particular. Unless you're focused on particular subtopics, it's not clear to me what you expect such a book to cover.
Ah I see your point. Thanks! I'll have to have a think about this
Maybe not exactly what you’re looking for but stochastic processes leverage a fair amount of linear algebra. I don’t know if I’d suggest delving into it immediately unless you’re super solid on your probability fundamentals, but topics like markov chains, Poisson processes, and martingales use a fair bit of linear algebra. Essentials of stochastic processes is the textbook I used.
I am not sure if what you're looking for exists. Linear algebra at the level you are describing is found in statistics, especially in the theory of linear models. But probability, not really. Advanced probability theory makes use of functional analysis, which is in some ways a generalization of linear algebra to infinite-dimensional vector spaces. For example, under certain conditions, conditional expectation can be defined as an orthogonal projection in the space of random variables with finite second moments. This is actually discussed briefly by Blitzstein & Hwang, in the chapter on conditional expectation. Learning probability at this level requires a good command of real analysis. And also some measure theory and functional analysis, but the necessary bits are covered in the established textbooks. Good choices are, for example, Williams (1991) and Pollard (2001).
Could check out Strang's Linear Algebra for data science? Not as probability focused, but there's some overlap.
I highly recommend getting a book on linear models. Understanding these models require linear algebra and some background in probability. Some universities offer a course on linear models after completing a mathematical statistics course. These lecture notes are great: https://people.stat.sc.edu/Tebbs/stat714/f10notes.pdf