Math 115AH Homework # 3

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 ) 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

and

be finite-dimensional vector spaces over the same field and let $\{e_{1},\dots ,e_{n}\}$ be a basis of

. 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 . Describe these functionals in terms of .

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

for any $A\in V$ is a linear transformation.

Problem 4 Let $V=\mathbb{C}$ be the set of all complex numbers. Recall that

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

which is linear over $\mathbb{R}$ (e.g., so called real-linear), but which is not a linear transformation of

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

which is complex-linear, but not real-linear).

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