MATH 3210 Class Notes 02-10-2025

Recall Linear Maps

Lemma

Suppose $v_{1}, \dots, v_{n}$ is a basis of $V$ and $w_{1}, \dots, w_{n} \in W$ . Then there exists a unique linear map $T : V \to W$ such that,

T v_{k} = w_{k} \forall k = 1, \dots, n .

Proof

All $v \in V$ can be written uniquely as $v = c_{1} v_{1} + \dots + c_{n} v_{n}$ . Define $T (c_{1} v_{1} + \dots + c_{n} v_{n}) = c_{1} w_{1} + \dots + c_{n} w_{n}$ …

Defined Algebra on the Set of Linear Maps

Definition

If $T \in L (V, W)$ and $S \in L (W, U)$ , then the product $ST \in L (V, U)$ is given by,

(ST) (v) = S (T v) .

Proposition

Assume all products “make sense”.

(Associativity) $(T_{1} T_{2}) T_{3} = T_{1} (T_{2} T_{3})$ .
(Identity) $T I_{V} = I_{W} T = T$
(Distributivity) $(S_{1} + S_{2}) T = S_{1} T + S_{2} T$ , and $S (T_{1} + T_{2}) = S T_{1} + S T_{2}$ .

Note: $ST \neq = TS$ .

A good example of this would be the “differentiation” and “multiplication by $x^{6}$ ” maps in the algebra of linear maps from $P (R)$ to itself. They very much so do not commute under composition.

Defined Nullspace

For the Exam

True/false conceptual questions.
Some computational problems (find the nullspace of a given map, determine a vector that is linearly independent with some set of vectors, etc.)
Some uniqueness proof (prove that the additive inverse is unique in a general vector space, etc.)
If it goes terribly it will likely be curved/adjusted.

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