Matrix Multiplication & Linear Transformations Flashcards

Question 1

What is the inner dimension compatibility rule for matrix multiplication?

Accepted Answer

To compute $C = AB$, the number of columns in $A$ must equal the number of rows in $B$.
If $A$ is $m 	imes k$ and $B$ is $k 	imes n$, the product $C$ exists and has dimension **$m 	imes n$**.

Question 2

What is the algebraic formula for entry $c_{ij}$ in matrix product $C = AB$?

Accepted Answer

$$c_{ij} = \sum_{k=1}^{p} a_{ik} b_{kj} = \mathbf{a}_{i, \ast} \cdot \mathbf{b}_{\ast, j}$$

Entry $(i, j)$ is the **dot product** of row $i$ of Matrix $A$ and column $j$ of Matrix $B$.

Question 3

Is matrix multiplication commutative ($AB = BA$)?

Accepted Answer

**No**. In general, $AB 
eq BA$.
Even when both are square matrices of size $n 	imes n$, matrix multiplication does not commute because the order of linear transformations matters.

Question 4

What is the arithmetic time complexity of multiplying two $n 	imes n$ matrices naively?

Accepted Answer

**$O(n^3)$ operations**.
Computing each of the $n^2$ entries requires a dot product of length $n$ ($n$ multiplications and $n-1$ additions).

Question 5

What is the geometric meaning of multiplying a 2D vector by a matrix $M$?

Accepted Answer

Multiplying $M\mathbf{v}$ applies a **linear transformation** (rotation, scaling, shear, reflection) to the space, mapping standard basis vectors $\hat{i} = [1, 0]^T$ and $\hat{j} = [0, 1]^T$ to the columns of $M$.

Question 6

What is the identity matrix $I$ and its multiplicative property?

Accepted Answer

A square matrix with $1$ on the main diagonal and $0$ elsewhere:
$$AI = IA = A$$
Multiplying by $I$ acts as the identity transformation, leaving any vector or matrix unchanged.

Question 7

What is the transpose of a matrix product: $(AB)^T$?

Accepted Answer

$$(AB)^T = B^T A^T$$
The order of multiplication is **reversed** when transposing a matrix product.

Question 8

Does matrix multiplication satisfy the associative property?

Accepted Answer

**Yes**. Matrix multiplication is strictly associative:
$$(AB)C = A(BC)$$
This allows regrouping multi-matrix multiplications for computational efficiency (e.g. Matrix Chain Multiplication).

Question 9

Given $A = \begin{pmatrix} 2 & 1 \ 0 & 3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \ 5 \end{pmatrix}$, compute $A\mathbf{v}$.

Accepted Answer

$$A\mathbf{v} = \begin{pmatrix} 2(4) + 1(5) \ 0(4) + 3(5) \end{pmatrix} = \begin{pmatrix} 8 + 5 \ 0 + 15 \end{pmatrix} = \begin{pmatrix} 13 \ 15 \end{pmatrix}$$

Question 10

If matrix $P$ is $4 	imes 2$ and matrix $Q$ is $3 	imes 5$, is $PQ$ defined?

Accepted Answer

**Undefined**. Inner dimensions do not match: $P$ has 2 columns while $Q$ has 3 rows ($2 
eq 3$). The dot product between rows of $P$ and columns of $Q$ cannot be formed.

Matrix Multiplication & Linear Transformations

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