Mathematics · Linear Algebra
2×2 Matrix–Vector Calculator
Apply a 2×2 linear transformation to a two-component vector.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- First component=2×4+0×5=8.
- Second component=0×4+3×5=15.
Problem → model → reason → result
What problem does this model solve?
Apply a 2×2 linear transformation to a two-component vector.
Why does the model apply?
Rows of the matrix dot with the input vector to produce transformed components.
What assumptions does it make?
Vector components share one coordinate basis, matrix entries are entered in the displayed positions, and the selected finite-dimensional model represents the transformation being studied.
Formula
y=Ax
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
[[2,0],[0,3]] times (4,5) gives (8,15).
Common mistake
Vector component order must match the matrix basis.
When does this model not apply?
Small real vectors and 2×2 matrices introduce the concepts but do not cover complex vector spaces, higher dimensions, numerical conditioning or every decomposition.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Matrix–vector product do?
Apply a 2×2 linear transformation to a two-component vector.
How does the Matrix–vector product work?
The calculator applies y=Ax. Rows of the matrix dot with the input vector to produce transformed components.
What can I learn from the Matrix–vector product?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.