Using Linear Algebra to Explain What a Perspective Is

Roles in Life and Linear Algebra

In everyday life, each role we take on comes with its own set of goals. Doctors aim to preserve life and heal the sick. Engineers design software, simulate system operations, and enhance efficiency through automation and bottleneck analysis. Entrepreneurs build organizations to create social impact while maximizing net profit.

Each role is driven by specific objectives and corresponding actions — these actions can be seen as input vectors. When these inputs pass through a transformation (represented by a matrix), they result in output vectors that reflect progress toward the goal.

So, what is a “perspective”?

Because each role has a different goal, the effectiveness of daily actions in moving toward that goal also varies. Behind every role lies a unique way of thinking — a mental model or theoretical framework that guides decision-making.

Linear algebra shows us that a matrix can be decomposed using Singular Value Decomposition (SVD) to identify a system’s orthogonal coordinate space. In this space, each axis (or eigenvector) reveals how much influence it has on the outcome (output vector), as measured by its corresponding eigenvalue (which can be thought of as a weight or impact factor). This tells us that, for any given system, there exists a set of directions (perspectives) along which progress toward the goal is most efficient.

In real life, each person’s perspective is like a unique coordinate system. Based on our individual goals, we extract and emphasize different elements from the same situation (input vectors). That’s why people in different roles can interpret the same event in very different ways.

Why should we embrace multiple perspectives?

Take the workplace as an example. A company includes many departments: sales, product, engineering, finance, HR. As an engineer, my daily focus is on software development, reducing technical debt, and improving team communication. In my mental framework, any module that could trigger a system failure stands out as a major issue — we naturally tend to amplify risks.

However, from a business or market perspective, the value of an engineer is not solely defined by technical skill. Even the most technically proficient engineer may be undervalued if their work is disconnected from business goals or market needs. This disconnect can affect both performance reviews and job interviews.

As we mature in our careers and our skills deepen, our goals tend to evolve. We aim to create more value and seek better returns. Yet, our daily habits and actions may still be influenced by our initial goals — this is what leads to blind spots.

Every framework that arises from career or skill development has its pros and cons. The downside is that it may trap us within a single level of perspective. Even when we are aware of a new goal, we still need a method to identify when our daily actions are misaligned with it.

For me, engaging in hobbies or having conversations with people from different professions is a great way to gain fresh insights and perspectives. It helps me see beyond my own framework and better align my actions with broader goals.