Summary of "Vector Space | Linear Transformation | Examples Of Linear Transformation | Linear Algebra"

Core definitions and concepts

Linear map / linear transformation (homomorphism): a mapping T : V → W between vector spaces over a field F is linear if and only if for all scalars a, b in F and all vectors α, β in V,
- T(α + β) = T(α) + T(β)
- T(a α) = a T(α)
- Equivalently: T(a α + b β) = a T(α) + b T(β) for all a, b ∈ F and α, β ∈ V.
Kernel (null space) of T: the set {α ∈ V : T(α) = 0}. The kernel is a subspace of V.
Range (image) of T: the set {T(α) : α ∈ V} ⊆ W. The range is a subspace of W.
Nullity: the dimension of the kernel, dim(ker T).
Rank: the dimension of the range, dim(range T). For a matrix, rank = dimension of the column space.

Two equivalent approaches:

Method A — direct combined test
- Take arbitrary vectors α, β ∈ V and scalars a, b ∈ F.
- Compute T(a α + b β).
- Simplify and show T(a α + b β) = a T(α) + b T(β).
- If this equality holds for arbitrary choices, T is linear.
Method B — separate tests
- Verify additivity: T(α + β) = T(α) + T(β) for all α, β.
- Verify homogeneity: T(c α) = c T(α) for all scalars c and vectors α.
- If both hold, T is linear.

Look for a nonzero constant term or any added constants (e.g., “+2”, “+3”) in the definition — that typically makes a map non-linear.
Look for nonlinear algebraic terms (e.g., x^2, y^2, products like xy) — these break linearity.
Produce a counterexample: pick simple vectors (standard basis vectors or 0) and scalars; show either T(α + β) ≠ T(α) + T(β) or T(c α) ≠ c T(α).

Find the set of possible outputs T(α). For linear maps, range = span{T(e1), T(e2), …} where {ei} is a basis of V.
Determine a basis for the image and its dimension (rank).

Take the standard basis vectors e1, e2, …, en of the domain.
Compute T(e1), T(e2), …, T(en).
Form the matrix whose columns are T(e1), T(e2), …, T(en). This matrix represents T relative to the standard bases.
Use the matrix to compute images of arbitrary vectors and to compute rank by column reduction if needed.

Worked proofs: show maps from R^3 → R^2 (or similar) are linear by applying the combined test — plug in coordinates, distribute scalars, and separate components.
Nonlinear examples: maps containing constant shifts (e.g., x + 2, y + 3) or squared terms fail the linearity tests; demonstrate failure by evaluating simple inputs.
Matrix example: to find the matrix of a componentwise-defined T on R^3, compute T(1,0,0), T(0,1,0), T(0,0,1) and place the results as columns to assemble the matrix.
Common exam tasks emphasized:
- Determine whether a mapping is linear.
- Compute kernel and range.
- Find a matrix representation.
- Compute rank and nullity.

Kernel and range are subspaces; nullity = dim(kernel), rank = dim(range).
Typical exam phrasing and common student pitfalls were highlighted.
Upcoming topics announced: rank of a matrix, Cayley–Hamilton theorem, and more exercises.
The presenter also advertised a second YouTube channel with General Aptitude material useful for NET preparation.