Linear Algebra Syllabus

Define the null space of a matrix $A$ and its dimension.

The null space is $\{\vec{x}:A\vec{x}=\vec{0}\}$, the solution set of the homogeneous system, a subspace of $\mathbb{R}^{n}$. Its dimension is the nullity of $A$.

Define the rank of a matrix and state its relationship to row and column spaces.

The rank is the common dimension of the row space and column space (they are always equal). It equals the number of nonzero rows (pivots) in the row echelon form of $A$.

State the Rank–Nullity Theorem for an $m\times n$ matrix $A$.

$\operatorname{rank}(A)+\operatorname{nullity}(A)=n$, where $n$ is the number of columns of $A$.

How do you find a basis for the column space and null space of $A$?

Row-reduce $A$. A basis for the column space consists of the columns of the original $A$ corresponding to pivot columns. A basis for the null space is obtained by solving $A\vec{x}=\vec{0}$ and writing the solution in terms of free variables (one basis vector per free variable).

What is the coordinate vector of $\vec{v}$ relative to a basis $B=\{\vec{v}_{1},\dots,\vec{v}_{n}\}$?

If $\vec{v}=c_{1}\vec{v}_{1}+\cdots+c_{n}\vec{v}_{n}$, then the coordinate vector is $[\vec{v}]_{B}=\begin{pmatrix}c_{1}\\ \vdots \\ c_{n}\end{pmatrix}$. The scalars are unique because $B$ is a basis.

Define the change-of-basis (transition) matrix $P$ from basis $B'$ to basis $B$.

$P=\left[\,[\vec{v}'_{1}]_{B}\;\cdots\;[\vec{v}'_{n}]_{B}\,\right]$, whose columns are the $B$-coordinates of the $B'$ basis vectors. It satisfies $[\vec{x}]_{B}=P\,[\vec{x}]_{B'}$.

How are the transition matrix from $B'$ to $B$ and from $B$ to $B'$ related?

They are inverses of each other: if $P$ maps $B'$-coordinates to $B$-coordinates, then $P^{-1}$ maps $B$-coordinates to $B'$-coordinates.

Define a linear transformation $T:V\to W$.

A function satisfying (1) $T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})$ and (2) $T(c\vec{u})=cT(\vec{u})$ for all $\vec{u},\vec{v}\in V$ and all scalars $c$. Equivalently $T(c\vec{u}+d\vec{v})=cT(\vec{u})+dT(\vec{v})$.

For a linear transformation $T$, what must $T(\vec{0})$ equal, and give an example of a non-linear map.

$T(\vec{0}_{V})=\vec{0}_{W}$ always. A map like $T(x)=x+1$ or $T(x,y)=(x^{2},y)$ is not linear because it fails additivity/homogeneity (and $T(\vec{0})\neq\vec{0}$ for the first).

Give examples of linear transformations on $\mathbb{R}^{2}$: rotation, projection, and the zero map.

Rotation by $\theta$: $T(\vec{x})=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}\vec{x}$; projection onto the $x$-axis: $T(x,y)=(x,0)$; zero transformation: $T(\vec{x})=\vec{0}$. Differentiation $D:P_{n}\to P_{n-1}$ is also linear.

Define the kernel of a linear transformation $T:V\to W$.

$\ker(T)=\{\vec{v}\in V:T(\vec{v})=\vec{0}\}$, a subspace of the domain $V$. Its dimension is the nullity of $T$.

Define the range of a linear transformation $T:V\to W$.

$\operatorname{range}(T)=\{T(\vec{v}):\vec{v}\in V\}$, a subspace of the codomain $W$. Its dimension is the rank of $T$.

State the dimension theorem (rank–nullity) for a linear transformation $T:V\to W$ with $\dim(V)=n$.

$\operatorname{rank}(T)+\operatorname{nullity}(T)=n$, i.e. $\dim(\operatorname{range}\,T)+\dim(\ker T)=\dim(V)$.

When is a linear transformation $T$ one-to-one (injective), in terms of its kernel?

$T$ is one-to-one if and only if $\ker(T)=\{\vec{0}\}$ (equivalently $\operatorname{nullity}(T)=0$).

When is a linear transformation $T:V\to W$ onto (surjective)?

$T$ is onto if and only if $\operatorname{range}(T)=W$, i.e. $\operatorname{rank}(T)=\dim(W)$.

For a linear transformation between spaces of equal finite dimension, how are one-to-one and onto related?

If $\dim(V)=\dim(W)$ is finite, then $T$ is one-to-one if and only if it is onto; either condition makes $T$ an isomorphism.

How is the standard matrix $A$ of a linear transformation $T:\mathbb{R}^{n}\to\mathbb{R}^{m}$ constructed?

$A=[\,T(\vec{e}_{1})\;T(\vec{e}_{2})\;\cdots\;T(\vec{e}_{n})\,]$, whose columns are the images of the standard basis vectors. Then $T(\vec{x})=A\vec{x}$.

How do you find the matrix of $T:V\to W$ relative to bases $B$ (domain) and $B'$ (codomain)?

$A=\left[\,[T(\vec{v}_{1})]_{B'}\;\cdots\;[T(\vec{v}_{n})]_{B'}\,\right]$, whose columns are the $B'$-coordinate vectors of the images of the $B$ basis vectors. Then $[T(\vec{x})]_{B'}=A\,[\vec{x}]_{B}$.

If $T$ has matrix $A$ relative to bases $B,B'$, how do you compute $[T(\vec{x})]_{B'}$?

$[T(\vec{x})]_{B'}=A\,[\vec{x}]_{B}$: convert $\vec{x}$ to $B$-coordinates, multiply by $A$, giving the $B'$-coordinates of $T(\vec{x})$.

If $T_{1}$ and $T_{2}$ are linear transformations with matrices $A_{1}$ and $A_{2}$, what is the matrix of the composition $T_{2}\circ T_{1}$?

The matrix product $A_{2}A_{1}$; that is, $(T_{2}\circ T_{1})(\vec{x})=A_{2}(A_{1}\vec{x})$. Composition corresponds to matrix multiplication (applied right to left).

When is a linear operator $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ invertible, and what is the matrix of $T^{-1}$?

$T$ is invertible if and only if its standard matrix $A$ is invertible ($\det(A)\neq 0$). Then the matrix of $T^{-1}$ is $A^{-1}$, and $T^{-1}(\vec{x})=A^{-1}\vec{x}$.

How does the matrix of an operator change under a change of basis, and what does this define?

If $A$ is the matrix relative to one basis and $P$ is the transition matrix, the matrix relative to the new basis is $A'=P^{-1}AP$. This defines similarity of matrices.

Define similar matrices and name three properties preserved by similarity.

$A$ and $B$ are similar if $B=P^{-1}AP$ for some invertible $P$. Similar matrices share the same determinant, trace, rank, characteristic polynomial, and eigenvalues.

How are matrix addition and scalar multiplication defined for $m\times n$ matrices?

Entrywise: $(A+B)_{ij}=a_{ij}+b_{ij}$ (matrices must have the same size) and $(cA)_{ij}=c\,a_{ij}$.

State the rule for matrix multiplication and the required dimension condition.

For $A$ of size $m\times n$ and $B$ of size $n\times p$, the product $AB$ is $m\times p$ with $(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}$. The number of columns of $A$ must equal the number of rows of $B$.

Is matrix multiplication commutative? State the associative and distributive laws.

Matrix multiplication is generally not commutative ($AB\neq BA$ in general). It is associative $(AB)C=A(BC)$ and distributive $A(B+C)=AB+AC$ and $(A+B)C=AC+BC$.

State the properties of the transpose: $(A+B)^{T}$, $(cA)^{T}$, $(AB)^{T}$, and $(A^{T})^{T}$.

$(A+B)^{T}=A^{T}+B^{T}$, $(cA)^{T}=cA^{T}$, $(AB)^{T}=B^{T}A^{T}$ (order reverses), and $(A^{T})^{T}=A$.

Define the inverse of a square matrix $A$ and give the $2\times 2$ formula.

$A^{-1}$ satisfies $AA^{-1}=A^{-1}A=I$. For $A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}$, $A^{-1}=\dfrac{1}{ad-bc}\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}$, provided $ad-bc\neq 0$.

State the rule for the inverse of a product and of a transpose.

$(AB)^{-1}=B^{-1}A^{-1}$ (order reverses) and $(A^{T})^{-1}=(A^{-1})^{T}$. Also $(A^{-1})^{-1}=A$ and $(cA)^{-1}=\frac{1}{c}A^{-1}$.

Define an elementary matrix and explain how it relates to row operations.

An elementary matrix is obtained from the identity $I$ by a single elementary row operation. Left-multiplying $A$ by that elementary matrix performs the same row operation on $A$. Every elementary matrix is invertible, and its inverse is also elementary.

Describe the Gauss–Jordan method for finding $A^{-1}$.

Form the augmented matrix $[\,A\mid I\,]$ and row-reduce to reduced row echelon form. If $A$ reduces to $I$, the result is $[\,I\mid A^{-1}\,]$; if $A$ cannot be reduced to $I$, then $A$ is not invertible (singular).