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Math 115AH Homework # 3

(additional problems)

Problem 1 (Direct Sums)   a) Show that $ \mathbb{R}^{n} $ is the direct sum of its subspaces $ W_{1} $ and $ W_{2} $ given by $ W_{1}=\{(a_{1},\dots ,a_{n})\in \mathbb{R}^{n}:\, a_{n}=0\} $ and $ W_{2}=\{(a_{1},\dots ,a_{n})\in \mathbb{R}^{n}:\, a_{1}=\dots =a_{n-1}=0\} $;

b) Show that the vector space of all upper-triangular matrices (i.e., matrices with the property that $ A_{ij}=0 $ for all $ i>j $) is the direct sum of the subspace $ W_{1} $ of diagonal matrices and the subspace $ W_{2}\doteq \{A\in M_{n\times n}(\mathbb{R}):\, A_{ij}=0\, \textrm{for all }i\geq j\} $.

c) Let $ V=M_{n\times n}(\mathbb{R}) $. Give an example (different from the example above) of two subspaces $ W_{1} $ and $ W_{2} $ such that $ V=W_{1}\oplus W_{2} $.

Problem 2   Let $ V $ and $ W $ be finite-dimensional vector spaces over the same field and let $ \{e_{1},\dots ,e_{n}\} $ be a basis of $ W $. Prove that any linear transformation $ T:V\to W $ is of the form

$ T(v)=\sum _{i=1}^{n}f_{i}(v)\cdot e_{i} $, where $ f_{i}\in V^{*} $ are some linear functionals on $ V $. Describe these functionals in terms of $ T $.

Problem 3   Let $ V=M_{n\times n}(\mathbb{R}) $ be the vector space of $ n\times n $ real matrices. Let $ B\in M_{n\times n}(\mathbb{R}) $ be a fixed matrix. Verify that $ T:V\to V $ defined by $ T(A)=AB-BA $ for any $ A\in V $ is a linear transformation.

Problem 4   Let $ V=\mathbb{C} $ be the set of all complex numbers. Recall that $ V $ can be considered as vector space over $ \mathbb{R} $ as well as a vector space over $ \mathbb{C} $. Give an example of a linear transformation of $ V $ which is linear over $ \mathbb{R} $ (e.g., so called real-linear), but which is not a linear transformation of $ V $ considered as a vector space over $ \mathbb{C} $ (i.e., is not so-called complex- linear). Is there an example of the opposite phenomenon (i.e., is there a transformation of $ V $ which is complex-linear, but not real-linear).

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Math 115AH Homework # 3

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Olga Radko 2003-08-04