Let C be an m × n matrix and let T : Rn → Rm be the linear transformation definedby C. Show that the matrix of T with respect to the standard bases of Rn and Rmis C
Question
Let C be an m × n matrix and let T : Rn → Rm be the linear transformation definedby C. Show that the matrix of T with respect to the standard bases of Rn and Rmis C
Solution
The matrix of a linear transformation T: R^n → R^m with respect to the standard bases of R^n and R^m is the matrix that represents the transformation. This matrix is obtained by applying the transformation to each vector in the standard basis of R^n and writing the result as a column in the matrix.
The standard basis for R^n is {e_1, e_2, ..., e_n}, where e_i is a vector with 1 in the i-th position and 0's elsewhere.
When we apply the transformation T to each standard basis vector e_i, we get the i-th column of the matrix C. This is because the transformation T is defined by the matrix C, so applying T to a vector is the same as multiplying the vector by the matrix C.
Therefore, the matrix of T with respect to the standard bases of R^n and R^m is the same as the matrix C. This shows that the matrix of a linear transformation defined by a matrix is the matrix itself.
Similar Questions
Let T be a linear transformation from an n dimensional vector space V to an mdimensional vector space W and let C be the matrix of T with respect to a basis Aof V and B of W . Show that(a) rank(T ) = rank(C);
If T : Rn → Rn is given by T (x1, x2, ..., xn) = (x2 + x3, x3, ..., xn, 0) then write downthe matrix of T w.r.t. the standard basis of Rn
et T and S be linear transformations on a finite dimensional vector space V, where S is non-singular. Then,ans.Rank (ST)= Rank(TS) ≠ Rank TRank (ST) ≠ Rank(TS) ≠ Rank TRank (ST) ≠ Rank(TS) = Rank TRank (ST) = Rank(TS) = Rank T Previous Marked for Review Next
Let T𝑇 and S𝑆 be linear transformations defined by T(x,y)=(2x−3y,−y)𝑇(𝑥,𝑦)=(2𝑥−3𝑦,−𝑦) and S(x,y)=(x−y,x)𝑆(𝑥,𝑦)=(𝑥−𝑦,𝑥). What is the matrix corresponding to the linear transformation T∘S𝑇∘𝑆?
(b) Assume that {v1, v2, · · · , vk} is a basis of Y . Show that the linear transform T : X → Y from (a) is invertible. (Hint: Use (a) to define a suitable linear transform U : Y → X and show that it is the inverse of T .)
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.