Matrix multiplication
Lecture 4
Auburn University
MATH 2660 - Spring 2026
January 14, 2026
Announcement
Michael Ofosu’s (our graduate TA) office has changed to Parker Hall 106 (previously 103). His office hours remain at the same time.
My office location and office hours are unchanged.
For the most up-to-date information, please refer to the course website.
Recap
$$ % Colors
% Coordinate vectors and matrices
% Common sets
% Abstract vector symbols
% Norms / absolute value
% Optional: dot product spacing (looks nicer in slides)
% Operators $$
Linear combinations and matrix notation
- An \(n\times k\) matrix represents \(k\) vectors in \(\mathbb{R}^n\).
- Each vector forms a column of the matrix: \[ \begin{bmatrix} \boxed{\begin{array}{c} v_{11} \\ \vdots \\ v_{n1} \end{array}} & \boxed{\begin{array}{c} v_{12} \\ \vdots \\ v_{n2} \end{array}} & \cdots & \boxed{\begin{array}{c} v_{1k} \\ \vdots \\ v_{nk} \end{array}} \end{bmatrix} \]
- We write this compactly as \([\vec{v}_1\ \vec{v}_2\ \cdots\ \vec{v}_k\)] (block notation).
- A linear combination of vectors \(\vec{v}_1,\dots,\vec{v}_k\) is a vector of the form \[ \vec{u} = c_1\vec{v}_1 + \cdots + c_k\vec{v}_k = \sum_{i=1}^k c_i\vec{v}_i, \] where the coefficients \(c_i\in\mathbb{R}\).
- If we encode the vectors into a matrix \(A=[\vec{v}_1\ \cdots\ \vec{v}_k]\) and the coefficients into a vector \[ \vec{c} = \begin{bmatrix} c_1 \\ \vdots \\ c_k \end{bmatrix}, \] then the linear combination can be written as a matrix multiplication \[ \vec{u} = A\vec{c}. \]
Get familiar with notational convention
\[ \begin{bmatrix} v_{\boxed{\phantom{\;\;}}} & v_{\boxed{\phantom{\;\;}}} & \cdots & v_{\boxed{\phantom{\;\;}}} \\ v_{\boxed{\phantom{\;\;}}} & v_{\boxed{\phantom{\;\;}}} & \cdots & v_{\boxed{\phantom{\;\;}}} \\ \vdots & \vdots & \ddots & \vdots \\ v_{\boxed{\phantom{\;\;}}} & v_{\boxed{\phantom{\;\;}}} & \cdots & v_{\boxed{\phantom{\;\;}}} \end{bmatrix} \begin{bmatrix} c_{\boxed{\phantom{\;\;}}} \\ c_{\boxed{\phantom{\;\;}}} \\ \vdots \\ c_{\boxed{\phantom{\;\;}}} \end{bmatrix} \\ = \begin{bmatrix} c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} + c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} + \cdots + c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} \\ c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} + c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} + \cdots + c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} \\ \vdots \\ c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} + c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} + \cdots + c_{\boxed{\phantom{\;\;}}}v_{\boxed{\phantom{\;\;}}} \end{bmatrix}. \]
Fill these in by using your conceptual understanding of matrix notation and matrix multiplication. (Hint: what is the meaning of column blocks? What is the meaning of \(v_{ij}\)?)
Matrix Multiplications
Coordinate system revisited
- In the Cartesian coordinate system, the coordinates \((3,-2)\) is interpreted as: “move 3 units in the \(\vec e_1\) direction, then move \(-2\) units in the \(\vec e_2\) direction.”
- What if we use a different coordinate system, with axes \[ \vec v_1 = \langle -1,0 \rangle, \qquad \vec v_2 = \langle 1,1 \rangle? \]
- Can you relate this to a linear combination?
- How can we visualize this movement?
Two coordinates of the same movement
Coordinate change
- We agree on the movement of \((3,-2)\) only after specifying a coordinate system.
- The movement is intrinsic, but its coordinate representation depends on the chosen system.
- When we write \[ \vec v_1 = \langle -1,0 \rangle, \qquad \vec v_2 = \langle 1,1 \rangle, \] these vectors are given in the Cartesian coordinate system.
- Once this information is known, we can convert coordinates from a new coordinate system into Cartesian coordinates.
- Matrix multiplication is exactly the tool that performs this conversion.
Matrix and coordinate change
- Interpret \(A = [\vec v_1 \ \vec v_2 \ \cdots \ \vec v_k]\) as the matrix encoding the axes of a new coordinate system (expressed in the Cartesian system).
- Let \(\vec u = \langle u_1,\dots,u_k \rangle\) be the coordinates of a point in this new system.
- Then \[\vec w = A\vec u\] gives the Cartesian coordinates of that point.
Coordinate change for multiple points
- In a fixed coordinate system, the axes stay the same, while different coordinates represent different points.
- Often, we want to convert many points at once from a new coordinate system into Cartesian coordinates.
- This can be done by placing multiple coordinate vectors side by side.
Matrix multiplication
Suppose we have \(m\) coordinate vectors \(\vec c_1,\dots,\vec c_m\). We collect them into a matrix: \[ [\vec c_1 \ \vec c_2 \ \cdots \ \vec c_m] = \begin{bmatrix} c_{11} & c_{12} & \cdots & c_{1m} \\ c_{21} & c_{22} & \cdots & c_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ c_{k1} & c_{k2} & \cdots & c_{km} \end{bmatrix}. \]
In the new coordinate system, the axes \(\vec v_1,\dots,\vec v_k\) are encoded by \[ [\vec v_1 \ \vec v_2 \ \cdots \ \vec v_k] = \begin{bmatrix} v_{11} & v_{12} & \cdots & v_{1k} \\ v_{21} & v_{22} & \cdots & v_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ v_{n1} & v_{n2} & \cdots & v_{nk} \end{bmatrix}. \]
Matrix multiplication (block notation)
- We define \[ [\vec v_1 \ \cdots \ \vec v_k] [\vec c_1 \ \cdots \ \vec c_m] = [\vec w_1 \ \cdots \ \vec w_m], \] where each column satisfies \[ [\vec v_1 \ \cdots \ \vec v_k]\vec c_\ell = \vec w_\ell. \]
Matrix multiplication (full expansion)
\[ \begin{bmatrix} v_{11} & v_{12} & \cdots & v_{1k} \\ v_{21} & v_{22} & \cdots & v_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ v_{n1} & v_{n2} & \cdots & v_{nk} \end{bmatrix} \begin{bmatrix} c_{11} & c_{12} & \cdots & c_{1m} \\ c_{21} & c_{22} & \cdots & c_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ c_{k1} & c_{k2} & \cdots & c_{km} \end{bmatrix} \\ = \begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1m} \\ w_{21} & w_{22} & \cdots & w_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ w_{n1} & w_{n2} & \cdots & w_{nm} \end{bmatrix}. \] with \(w_{i\ell} = c_{1\ell}v_{i1} + c_{2\ell}v_{i2} + \cdots + c_{k\ell}v_{ik} = w_{i\ell} = \sum_{j=1}^k c_{j\ell}\, v_{ij}.\)
- This is the general matrix multiplication, extending the earlier case of multiplying a matrix by a single vector.
- Interpretation of each matrix:
- \(n\times k\): axes of the new coordinate system
- \(k\times m\): coordinates of \(m\) points in that system
- \(n\times m\): Cartesian coordinates of the same points
- \(n\times k\): axes of the new coordinate system
- Dimension check: \((n\times k)\cdot(k\times m) \Rightarrow (n\times m)\), so matrix multiplication is defined precisely when the middle dimensions match.
- Interpretation:
- if you have \(k\) axes, then each point must be described using \(k\) coordinates.
- Since each axis is a vector in \(\mathbb{R}^n\), multiplying by \(m\) such coordinate vectors produces \(m\) Cartesian points in \(\mathbb{R}^n\).
Example
- Consider the coordinate system with two axes \[ \vec v_1=\langle -1,0 \rangle, \qquad \vec v_2=\langle 1,1 \rangle. \]
- In this system, consider four points with coordinates \[ (3,-2), \quad (0,1), \quad (2,0), \quad (1,1) \]
- Use one matrix multiplication to find the Cartesian coordinates of all four points.
- Type your answer in the form \((x_1,y_1), (x_2, y_2), (x_3,y_3), (x_4,y_4)\).
Scan the QR code or go to join.iclicker.com/MBNJ.
Note: If you are completely lost, or facing any difficulties, please briefly explain your situation instead.
Computer-aided calculations
- Many exercises in this course are designed to build your eyeball and muscle memory for new concepts, rather than to practice perfectly error-free calculations.
- In practice, computers and calculators are often more reliable than humans, and knowing how to use them effectively is an important skill.
- For the best learning experience, I recommend first working through exercises by hand to develop understanding (which is much more effective than simply reading worked-out solutions), and then checking your results using a computer.
MATLAB
- One widely used computational tool in engineering and applied sciences is MATLAB.
- You may install MATLAB on your laptop, or use the web version on any device.
