# Linear transformations and subspaces

B1) This question concerns the following two subsets of :

(a) Show that , and find a vector in that does not belong to T. [3]

(b) Show that T is a subspace of . [4]

(c) Show that S is a basis for T, and write down the dimension of T. [7]

(d) Find an orthogonal basis for T that contains the vector . [3]

(e) Express the vector of T as a linear combination of the vectors in your orthogonal basis for T. [3]

B2 This question concerns the function t given by the rule

(a) Use the strategy in Unit 4, section 1, to show that t is a linear transformation. [3]

(b) Write down the matrix for t with respect to the standard basis in both the domain and codomain. [1]

(c) Determine the matrix of t with respect to the domain basis S of and the standard basis in the codomain. [2]

(d) Determine the matrix of t with respect to the domain basis S of basis and codomain basis of U. [5]

(e) Find the kernel of t, and states its dimension. [3]

(f) Let be a linear transformation given by . Determine the matrix of m and with respect to the standard basis in both the domain and the codomain. [6]

Please see the attached file for the fully formatted problems.

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#### Solution Summary

Linear transformations and subspaces are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.