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Chapter 4 Review

Theorem: Let $S=\{a_1,\ldots,a_m\}$ be a set of vectors of $\mathbb{R}^n$. Let $A=[a_1\; \ldots \; a_n]$ be a matrix and $T:\mathbb{R}^m \to\mathbb{R}^n$ be the linear transform defined by $T(x)=Ax$. Let $B$ be an echelon form of $A$. Then the following objects are equal:

• The set of vectors killed by $T$,

• $\{x:Ax=0\}$ (this is the set of homogeneous solutions to $A$),

• null($A$),

• $\{x:T(x)=0\}$,

• ker($T$),

• number of rows of all zeros in $B$,

• The set of vectors hit by $T$,

• $\{T(x): x\in \mathbb{R}^n\}$,

• range($T$),

• col($A$),

• span($S$),

• dim(col($A$)),

• dim(range($T$)),

• dim(span($S$)),

• $m$ - nullity($A$) (rank-nullity theorem),

• $m$ - dim(ker($T$)),

• dim(row($A$)), (think of this as maximal number of linear independent equations in $Ax=0$),

• number of pivots in $B$,

Example: Let $T(x)=Ax$, where $A$ is

$\begin{bmatrix} 1 & 2 & 0 & 2 \\ -2 & -4 & 1 & -3 \\ 1 & 2 & 2 & 4 \end{bmatrix}$

and has reduce echelon form $B$ given by

$\begin{bmatrix} 1 & 2 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

• What is the range of $T$?

• What is the kernel of $T$?

• What is the row space of $A$?

• What is the rank of $A$?

• What is the nullity of $A$?

• Write the columns corresponding the free variables as a linear combination of the pivot columns.

• What is the general solution to $Ax=0$?

• What is the general solution to $Ax=[2-3,4]^t$?

• What is a vector not in the range of $T$?

Example: Answer all the same questions as above but for an invertible transform.

Example: Give an example of a linear transform $T:\mathbb{R}^3 \to \mathbb{R}^2$ such that $T(1,1,0)=(1,0)$ and $T(0,1,2)=(1,2)$.

• What is the smallest possible rank such an example could be?

• What is the largest possible rank such an example could be?

• What is the smallest possible nullity such an example could be?

• What is the largest possible nullity such an example could be?