Linear Algebra · Book or PDF

A First Course in Linear Algebra

A.G. Hamilton · Study material generated by AI from this edition. Not affiliated with the publisher.

Cheat sheet

Key terms

Term	Meaning
Linear equation	Each unknown appears only to the first power, with no products of unknowns ( $2x+3y=5$ is linear; $xy=4$ is not)
Elementary row operations	Swap two rows; multiply a row by a non-zero number; add a multiple of one row to another
Row-echelon form	Each row starts with a leading 1 that sits right of the leading 1 above it; zero rows go at the bottom
Inconsistent	No solution exists; a reduced row $(0\ \dots\ 0\,\vert\,1)$ means $0=1$
Augmented matrix $[A\vert h]$	Coefficient matrix with the right-hand sides added as a last column
Rank	Number of non-zero rows left after Gaussian elimination (GE)
Transpose $A^T$	Rows become columns; the $(i,j)$ -entry is $a_{ji}$
Symmetric / skew-symmetric	$A^T=A$ / $A^T=-A$ (a skew matrix has a zero diagonal)
Orthogonal matrix	$A^TA=AA^T=I$
Triangular / diagonal	Zeros below (upper) or above (lower) the diagonal / zeros everywhere off the diagonal
Elementary matrix	$I$ changed by one row operation
Invertible / singular	Some $A^{-1}$ gives $AA^{-1}=A^{-1}A=I$ / no inverse exists
Linear combination	$x_1v_1+\dots+x_kv_k$
LD / LI	Linearly dependent: some combination with not all coefficients 0 gives $\mathbf 0$ . Linearly independent: only all-zero coefficients give $\mathbf 0$
Cofactor $A_{ij}$	Signed minor: delete row $i$ and column $j$ , take the determinant, then apply the $\pm$ checkerboard
Adjoint adj $A$	Transpose of the matrix of cofactors
Position vector $\mathbf r_A$	$\overrightarrow{OA}$ ; its components are the coordinates of A
Normal vector	A vector perpendicular to a plane

Formulas and rules

Matrices

Product: $(AB)_{ij}=\sum_k a_{ik}b_{kj}$ . A $p\times q$ matrix times a $q\times r$ matrix gives a $p\times r$ matrix.
Associative and distributive: $(AB)C=A(BC)$ and $A(B+C)=AB+AC$ . But in general $AB\neq BA$ .
Transpose rules: $(A^T)^T=A$ , $(A+B)^T=A^T+B^T$ , $(AB)^T=B^TA^T$ .
Inverse of a product: $(AB)^{-1}=B^{-1}A^{-1}$ .
$2\times2$ inverse: $\begin{bmatrix}a&b\\c&d\end{bmatrix}^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ , provided $ad-bc\ne0$ .
Inverse by cofactors: $A^{-1}=\frac{1}{\det A}\,\text{adj}\,A$ . Product rule: $\det(AB)=\det A\det B$ .

Determinants

$\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc$ .
The $3\times3$ sign pattern is $+-+/-+-/+-+$ .
Triangular matrix: $\det$ is the product of the diagonal entries.
Effect of row operations: a swap changes the sign; scaling a row by $k$ multiplies $\det$ by $k$ ; adding a multiple of a row leaves $\det$ unchanged.
Two equal rows give $\det=0$ .

Solving $Ax=h$

Condition	Outcome
rank $[A\vert h]>$ rank $A$	No solution
Both ranks $=q$ (the number of unknowns)	Unique solution
Both ranks $=r<q$	Infinitely many solutions, with $q-r$ parameters
$A$ square and invertible	$x=A^{-1}h$

Equivalence Theorem ( $p\times p$ matrix $A$ ). All of these are equivalent:

$A$ is invertible;
rank $A=p$ ;
the columns of $A$ are LI;
$Ax=0$ has only the solution $x=0$ ;
$\det A\ne0$ .

Vectors and geometry

Triangle Law: $\overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR}$ . Also $\overrightarrow{AB}=\mathbf r_B-\mathbf r_A$ .
Section Formula (P divides AB in ratio $m:n$ ): $\mathbf r_P=\frac{n\mathbf r_A+m\mathbf r_B}{m+n}$ . Centroid: $\frac13(\mathbf a+\mathbf b+\mathbf c)$ .
Length and unit vector: $|x\mathbf i+y\mathbf j+z\mathbf k|=\sqrt{x^2+y^2+z^2}$ ; unit vector $=\frac{1}{|\mathbf a|}\mathbf a$ .
Line: $\mathbf r=\mathbf a+t\mathbf u$ , or through two points $\mathbf r=\mathbf a+t(\mathbf b-\mathbf a)$ .
Plane: $(\mathbf r-\mathbf a)\cdot\mathbf n=0$ , which expands to $ax+by+cz+d=0$ with $\mathbf n=(a,b,c)$ .
Dot product: $\mathbf a\cdot\mathbf b=a_1b_1+a_2b_2+a_3b_3=|\mathbf a||\mathbf b|\cos\theta$ . A value of 0 means the vectors are perpendicular.
Distance from a point to a plane: $\dfrac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}$ .
Cross product: $\mathbf a\times\mathbf b=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$ , with $|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta$ .
Areas: triangle $=\frac12|\mathbf a\times\mathbf b|$ ; parallelogram $=|\mathbf a\times\mathbf b|$ .
Volume of a parallelepiped: $|\mathbf a\cdot(\mathbf b\times\mathbf c)|$ . A value of 0 means the vectors are coplanar (LD).

Processes

Gaussian elimination
1. Scale row 1 so its first entry is 1.
2. Clear the entries below that leading 1.
3. Repeat down the rows.
4. If a pivot is 0, swap with a lower row that has a non-zero entry there. If none exists, skip to the next column.
5. Back-substitute from the bottom up.
Classify a system
1. Reduce $[A\vert h]$ .
2. A row $(0\dots0\vert1)$ means no solution.
3. Otherwise, set each unknown without a leading 1 equal to a parameter ( $t, u,\dots$ ).
4. If the system contains a parameter $c$ , check separately every value of $c$ that makes a pivot 0.
Find $A^{-1}$
1. Write $[A\vert I]$ .
2. Row-reduce until the left side becomes $I$ ; the right side is then $A^{-1}$ .
3. If a zero row appears on the left, $A$ is singular.
Test vectors for LI
1. Put the $q$ vectors in as columns and apply GE.
2. $q$ non-zero rows means LI; fewer means LD.
Large determinant
1. Apply GE and record the effect of each operation on $\det$ .
2. Multiply the diagonal entries of the triangular result.
3. Undo the recorded changes.
Plane through A, B, C
1. Take $\mathbf n=\overrightarrow{AB}\times\overrightarrow{AC}$ .
2. Use the point-normal form with A.
Do two lines meet?
1. Give each line its own parameter.
2. Equate x, y and z, then solve the 3 equations in 2 unknowns.
3. If the system is inconsistent, the lines do not meet.

Watch out

Wrong belief	Correction
$AB=BA$ , so $(A+B)^2=A^2+2AB+B^2$	Order matters: $(A+B)^2=A^2+AB+BA+B^2$
$(AB)^{-1}=A^{-1}B^{-1}$	Reverse the order: $B^{-1}A^{-1}$ (same for transposes)
A product of symmetric matrices is symmetric	$(AB)^T=BA$ , which is usually $\ne AB$
Fewer equations than unknowns means infinitely many solutions	It can also be inconsistent, with no solution
More equations than unknowns means no solution	Extra equations may be redundant
No zero column means invertible	Check the rank or $\det$ . Also, rank $\le\min(p,q)$
A determinant is a matrix	$\det A$ is a single number computed from a square matrix
One line has one set of parametric equations	Any multiple of the direction vector and any point on the line give the same line
$\mathbf a\times\mathbf b=\mathbf b\times\mathbf a$	$\mathbf a\times\mathbf b=-\mathbf b\times\mathbf a$ ; also $\mathbf a\times\mathbf a=\mathbf 0$
The cross product works in 2D	It is defined only for 3-vectors
The Sarrus diagonal trick works for any size	It works only for $3\times3$

Mnemonics

Socks and shoes: you undo the last thing first, so $(AB)^{-1}=B^{-1}A^{-1}$ and $(AB)^T=B^TA^T$ .
Dominoes: $(p,q)\cdot(q,r)\to(p,r)$ . The middle numbers must match.
"I Really Can't Have Determinant zero": Invertible, Rank $p$ , Columns LI, Homogeneous system has only $x=0$ , $\det\ne0$ . These stand or fall together.
None, One, Lots: every linear system has exactly one of these three outcomes.
Cross product clock i→j→k→i: going forward around the clock gives $+$ ( $\mathbf i\times\mathbf j=\mathbf k$ ); going backward gives $-$ ( $\mathbf j\times\mathbf i=-\mathbf k$ ).

A First Course in Linear Algebra

Cheat sheet

A First Course in Linear Algebra: cheat sheet

Key terms

Formulas and rules

Processes

Watch out

Mnemonics

Cheat sheet

A First Course in Linear Algebra: cheat sheet

Key terms

Formulas and rules

Processes

Watch out

Mnemonics