## Fundamental Theorem of Linear Maps - T : V $\arrow$ W ; linear map in sense of set theory - compatible with definitions of addition, scalar multiplicaton - Define two subspaces with this linear map, one in V, one in W - In V: - define Null space in V: Null T - subset of V: elements that T maps to 0 - $\{ v \in V\ \mid T(v) = 0_W\}$ - subset of V of v's; condition describing those elements - here, the v's mapped to 0, the 0 in W - - Lemma 1: Null T is subspace of V: - Proof: simple criterion: 3 conditions: 0, closed under addition; closed under scalar multiplication; - T sends 0 of V to 0 of W - v,w then sum T(v) + T(w); v,w in null space = 0 + 0 - prove - Homeworks of last week - 2C, 3A - Example: **null space** ; R2 to R; (x,y) to x-y=0 - if equals 1, not subspace; does not contain 0 - affine space: translation down by 1; not subspace - homogeneous - Example: homework considered subspace U of P4 so all elements so D'' = 6 = 0 - Exercise - from P4 to P''; - interpret as null space over linear map; from P4 to P2 at point 6 - evaluate sum at point 6 is sum of values - derivative followed by derivative followed by evaluation at 6 is composition of 3 linear maps - null space is U; those polynomials that T takes to 0; not 0, not all things - dimension of null space is 4, one less than 5 - take P(x) to x P(x); those that multiplied by x go to 0; so is 0 polynomial; as small as possible - other HW last week - Example: T of P(F) polynomials of all degrees; - product of two functions that is 0 - Def: second subspace: first subset of W, then show it's a subspace: **Range**, is subset of W that are TV for some v in V: - image - Lemma 2: range is subspace of V - 0 in range? yes, 0 of V is mapped to 0 of W - Ideas from Set theory, not linear algebra; - two sets, map between, see image as subset; - could be element z; preimage is in S1 that maps to z - injective map or 1-1; not good; use injective; 1-1 map not same as 1-1 correspondence - If f y1 - surjective: all in W - injective and surjective is bijective - check that Linear map is injective - check that linear map is surjective and injective using ideas of null space and range - Lemma 3: injective if and only if null space consists of {0} - Lemma 4: surjective if range is all of W - Th: Suppose V is fin dim VS over F; and T: V - W; then Dim V = Dim null of T + dim range of T - How would we prove ?: 53:00: - Start with basis of the null space, which is a subspace of V - What is the basis of a zero vector space? - Empty set is the basis of the zero set - Extend basis of null space to basis of entire space V --- at end deal with Null T is {0} the notion of a basis includes two properties: LI, spanning huge improvement with finite dim V,, know a priori what the dimension is checking LI is easier; but if know dimension, don't need to prove other how do you know dimension p we know consider P 4;