Thm 2.4: Let T:V→W be linear. Then T is one-to-one ⇔ N(T)={0_V}.

Rank and Nullity of a linear transformation described with several examples.

Types of Linear Transformation: Identity, Zero, Injective(one-to-one), Bijective(onto) & Surjective.

Thm 2.5 : Let T:V→W be linear. Equivalent statements: T is one-to-one ⇔ T is onto ⇔ rank(T)= dim(V).

Ex 10: Determine the dim(R(T)) ; where T:P2(R)→M(2×2)(R) and T(f(x)) = ((f(1)-f(2) & 0 @ 0 & f(0))).

Ex 11: Determine whether T is one-to-one or onto; T:P2(R)→P3(R) and T(f(x))=2f'(x)+∫_0^x (3f(t))dt.

Thm 2.2: Let T: V → W be linear. If β = {v1,v2, … ,vn } is a basis for V, then R(T) = span(T(β)).

Thm 2.1: Let T: V→W be linear. Prove that N(T) and R(T) are subspaces of V and W, respectively.

(Null space & Range): Null space/kernel and range/image are explained in detail with many examples.

(Dimension/Rank-Nullity Theorem): Statement and proof have been explained in detail to all students.

Linear Transformation Examples: Explained 5 examples with step-by-step explanations.

Linear Transformation: A complete guide to linear transformations, with definitions and examples.

Corollary 2 (Replacement Theorm): An independent subset extends (generating set reduced) to a basis.

Corollary 2 (Replacement Thm): Number of vectors in a generating (independent) set and become basis.

34. Prove that if W1 is any subspace of V, then there exist a subspace W2 of V s.t. V = W1 ⊕ W2.

Grassmann's Theorem: For subspaces W1, W2 ⊆ V: dim(W1 + W2) = dim(W1 ) + dim(W2) - dim(W1 ∩ W2).

12*. If {v1 ,..., vn} a basis ⇒ {v1, v2 - v1, v3 - v1, ... ,vn - v1} is also a basis for V.

12. Let β = {u,v,w} is a basis for V. Show that γ = {u+v+w, v+w, w} is also a basis for V.

Thm 1.11: For subspace W⊆V: (a) W is finite-dimensional & dim(W) ≤ dim(V). (b) dim(W) = dim(V)⇒V=W.

Corollary 1: Every basis of a vector space V contains the same number of vectors.

Thm 1.10 (Replacement Theorem): Statement of the replacement theorem (Steinitz Exchange Theorem).

Thm 1.8: Set β is a basis for V ⇔ each v ∈ V can be uniquely expressed as lin. com. of vectors in β.

(Dimension): What is the dimension of a vector space? Explained with the definition and examples.

(Basis): What is Basis of a vector space? Explained with a lot of examples.

Thm 1.7: Let 𝑆 ⊆ 𝑉 inde. & 𝑣 ∈ 𝑉 that is not in 𝑆. Then 𝑆 ∪ {𝑣} is linearly dependent ⇔ 𝑣 ∈ span(𝑆).

(Independent & Dependent): Definition and examples of linearly independent & dependent subsets of V.

(Sigma Algebra): What is σ-Algebra is measure theory? Discussed with definition and examples.

(Algebra): What is Algebra is measure theory? Discussed with definition and several examples.

(Span & Generate): Defining "Span of S" and "Generating V" with examples (explained key differences)

15. Show that if S1, S2 ⊆ V then span(S1 ​∩ S2) ⊆ span(S1​) ∩ span(S2​). Also, give two examples.

14. If S1 and S2 are nonempty subsets of V, then span(S1 ∪ S2 ) = span(S1 ) + span(S2 ).

12. For W ⊆ V, Show that W is a subspace of V ⇔ span(W) = W. For S ⊆ V, span(span(S)) = span(S).

Thm 1.5: For S⊆V, span(S) is a subspace of V & span (S) is contained in every subspace containing S.

(Linear Combination): What is a Linear Combination? Full Concept with Easy Examples.

26. Prove that V is the direct sum of upper triangular and strictly lower triangular matrices.

23. Show: (a) W1 + W2 is a subspace of V. (b) W1 + W2 smallest subspace of V containing W1 and W2.

19. Prove that W1 ∪ W2 is a subspace of V ⇔ W1 ⊆ W2 or W2 ⊆ W1.

Thm 1.4: Any intersection of subspaces of a vector space V is also a subspace of V.

Thm 1.3: W is a subspace of V ⇔ (i) 0_V ∈ W ; (ii) x + y ∈ W (iii) cx ∈ W ;x, y ∈ W, c ∈ F.

(Vector subspace): What is a Vector Subspace? Full Explanation with Properties and Examples

Thm 1.2: (a) (zero times a vector): For every x∈V, 0∙x=0_V. (b) For every c∈F, c∙0_V=0_V.

Thm 1.1 (Corollary 1, 2): A vector space has a unique additive identity and unique additive inverse.

Thm 1.1 (Cancellation law for vector addition): If x, y, z ∈ V such that x + z = y + z then x = y.

(Vector Space): What is a Vector Space? Full Concept with Properties & Examples.

Field in Abstract Algebra: Definition, Properties & Examples | Complete Concept.

(Topology): Topology in Real Analysis and Measure Theory | Key Concepts Explained

Test 2: Trigonometric identity + Double-angle formulas + Solve different Trigonometric Equations.

Test 3 (Law of Sines & Cosines): Solve Any Triangle and Find Its Area.

Test 4 (Rectangular ↔ Polar Coordinates): Point/Equations conversion tutorial.

(Forgot password reset): How to reset the password of Macbook/ iMac. #macbook #macbookpro #imac

iCam365 app: Full details (app install, account create, add camera, insert SD card, reset it).

How to combine several PDF files into a single one.

How to factory reset a Samsung Chromebook. #chromebooks #chromebook

TP-Link Tapo app: How to add Tapo/TP-Link security camera (shows all steps). #securitycamera

iCam365 camera setup (Explained A to Z): How to add camera to iCam365 app.

iCam365 app (shows all steps): How to set up a security camera in the iCam365 app. #securitycamera

Baby Balance Bike Assembly: DRAMATION Baby Balance Bike.

Young's inequality: Let p,q ∈(1,∞) s.t. 1/p+1/q=1. Then for any a,b∈(0,∞), ab ≤ a^p/p + b^q/q.

Lemma: If p∈ [1,∞) and a,b \ge 0, then (a+b)^p \le 2^(p-1) (a^p+b^p).

1. Solve the PDE: u_x+u_y=u^2 ; u(x,0)=h(x) using the method of characteristics.