Range and Column Space 2

For a linear transformation T: Rⁿ → R^m given by an m by n matrix A = [a₁ a₂ ... a_n], we have (see this exercise for a more concrete example)

Thus rangeT is given by all the linear combinations (i.e., for all choices of the coefficients x₁, x₂, ..., x_n) of the columns of A.

The column space of a matrix A is

colA = {all the linear combinations of columns of A}
= {Ax: any x ∈ Rⁿ}
= {b ∈ Rⁿ: Ax = b has solutions}.

Example Because

(a + b, 3a - b, 2a + b) = a(1, 3, 2) + b(1, -1, 1),

the subspace

{(a + b, 3a - b, 2a + b): a, b ∈ R} ⊂ R³

given in an earlier example is in fact the columns space of the matrix

`A` = [	1	1	].
	3	-1
	2	1

Notice that the linear transformation in the earlier example is also given by the matrix.

We also note that since the system Ax = b has solutions for b = (3, 1, 4) and has no solution for b = (1, 0, 1), the vector (3, 1, 4) is in colA, and the vector (1, 0, 1) is not in colA.

`T`(`x`) = [	`a`₁₁`x`₁ + `a`₁₂`x`₂ + ... + `a`_1n`x_n`						]
	`a`₂₁`x`₁ + `a`₂₂`x`₂ + ... + `a`_2n`x_n`
	:
	`a`_m1`x`₁ + `a`_m2`x`₂ + ... + `a_mnx_n`

=	`x`₁[	`a`₁₁	] + `x`₂[	`a`₁₂	] + ... + `x_n`[	`a`_1n	]
		`a`₂₁		`a`₂₂		`a`_2n
		:		:		:
		`a`_m1		`a`_m2		`a_mn`

=	`x`₁`a`₁ + `x`₂`a`₂ + ... + `x_na_n`

Range and Column Space

2. Column Space

[exercise]
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Range and Column Space

2. Column Space

[exercise] [previous topic] [part 1] [part 2] [next topic]

[exercise]
[previous topic] [part 1] [part 2] [next topic]