So I know that matrix multiplication is not commutative. And yet, I am taught that transformation matrices can be composed to arrive at a single transformation. So then, suppose that I have 3 matrices - each representing a rotation about each axis of a 3D coordinate system. How can I be sure that I arrive at the correct result, if the order matters?
2026-03-30 11:52:36.1774871556
Combining matrix transformations
844 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in MATRICES
- How to prove the following equality with matrix norm?
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- Powers of a simple matrix and Catalan numbers
- Gradient of Cost Function To Find Matrix Factorization
- Particular commutator matrix is strictly lower triangular, or at least annihilates last base vector
- Inverse of a triangular-by-block $3 \times 3$ matrix
- Form square matrix out of a non square matrix to calculate determinant
- Extending a linear action to monomials of higher degree
- Eiegenspectrum on subtracting a diagonal matrix
- For a $G$ a finite subgroup of $\mathbb{GL}_2(\mathbb{R})$ of rank $3$, show that $f^2 = \textrm{Id}$ for all $f \in G$
Related Questions in LINEAR-TRANSFORMATIONS
- Unbounded linear operator, projection from graph not open
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- A different way to define homomorphism.
- Linear algebra: what is the purpose of passive transformation matrix?
- Find matrix representation based on two vector transformations
- Is $A$ satisfying ${A^2} = - I$ similar to $\left[ {\begin{smallmatrix} 0&I \\ { - I}&0 \end{smallmatrix}} \right]$?
- Let $T:V\to W$ on finite dimensional vector spaces, is it possible to use the determinant to determine that $T$ is invertible.
- Basis-free proof of the fact that traceless linear maps are sums of commutators
- Assuming that A is the matrix of a linear operator F in S find the matrix B of F in R
- For what $k$ is $g_k\circ f_k$ invertible?
Related Questions in COORDINATE-SYSTEMS
- How to change a rectangle's area based on it's 4 coordinates?
- How to find 2 points in line?
- Am I right or wrong in this absolute value?
- Properties of a eclipse on a rotated plane to see a perfect circle from the original plane view?
- inhomogeneous coordinates to homogeneous coordinates
- Find the distance of the point $(7,1)$ from the line $3x+4y=4$ measured parallel to the line $3x-5y+2=0.$
- A Problem Based on Ellipse
- Convert a vector in Lambert Conformal Conical Projection to Cartesian
- Archimedean spiral in cartesian coordinates
- How to find the area of the square $|ABCD|$?
Related Questions in ROTATIONS
- Properties of a eclipse on a rotated plane to see a perfect circle from the original plane view?
- why images are related by an affine transformation in following specific case?(background in computer vision required)
- Proving equations with respect to skew-symmetric matrix property
- Finding matrix linear transformation
- A property of orthogonal matrices
- Express 2D point coordinates in a rotated and translated CS
- explicit description of eigenvector of a rotation
- Finding the Euler angle/axis from a 2 axes rotation but that lies on the original 2 axes' plane
- How to find a rectangle's rotation amount that is inscribed inside an axis-aligned rectangle?
- Change of basis with rotation matrices
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
You are conflating two different things. Given two matrices $A,B$ it is true that the order in which list them before you multiply them matters in the generic case - you wouldn't expect $AB = BA$ to be true without further information. The technical statement of this is that matrix multiplication may not be commutative.
When you have 3 or more matrices we can talk about a different kind of "order". If you multiply the three matrices to get the product $ABC$ the question arises whether this means you multiply $A$ and $B$ to get another matrix $X$ and then multiply $X$ and $C$. Or should you multiply $B$ and $C$ to get another matrix $Y$ and then multiply $A$ and $Y$. The fact is, out of those two possibilities you always get the same result. It is in this sense that the "order doesn't matter" which you are talking about. The technical name for this is that matrix multiplication is associative.