The Range as Image

Linear Algebra (Matrices) · Axiom Academy

LESSON The Image of a Transformation The image of T is everything it can reach — and that set is exactly the column space of its matrix. 1. The Image Is the Span of the Columns Watch the columns of A do the work. Feeding T the basis vectors gives and — the columns themselves. Every other output is a combination of those two columns, so as ranges over the domain the outputs sweep out the entire span of the columns . The image is the set of all outputs …which is exactly the column space of A 2. When the Image Collapses to a Line The image is not always the whole codomain. If the columns of A are parallel , they span only a line — and no matter which input you pick, every output lands on that one line. Watch the entire input plane get crushed onto a single direction. 3. Onto Means the Image Fills the Codomain Put the two cases side by side. T is surjective ( onto ) when its image equals the entire codomain — every target is hit. For this happens exactly when , so the columns span all of . Watch the full-rank map flood the codomain while the rank-deficient one leaves everything off its line unreached. Worked check — the two matrices above, side by side. Full rank : the columns are independent, so Rank deficient : the columns are parallel, so Here , so T is not onto — every point off the line y_2=2y_1 is unreachable. . The image is a proper subspace (a line here); some outputs can never be produced.

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