how to find the range of a matrix transformation

Let us consider three matrices X, A and B such that X = AB. The image of T is the x1¡x2-plane in R3. (3pt) Verify that the matrix defines a linear transformation that is one-to-one and onto. Suppose there exist vectors {→a1, ⋯, →an} in Rn such that (→a1 ⋯ →an) − 1 exists, and T(→ai) = →bi Then the matrix of T must be of the form (→b1 ⋯ →bn)(→a1 ⋯ →an) − 1. The range of a linear transformation L from V to W is a subspace of W. Proof. It turns out that the matrix A of T can provide this information. A. ker(T). L ( v1) = w1 and L ( v2 ) = w2. The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation). Remarks I The range of a linear transformation is a subspace of its codomain. Lesson 22 Domain and Range of a Transformation 3 Example 4: [Let =( ) be a function with domain =−6,5]and range =[0,14]. This means that range (T)= Col (A) is a subspace of R m of dimension less than m : perhaps it is a line in the plane, or a line in 3 -space, or a plane in 3 -space, etc. In other words, a linear transformation T:

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