Section II.2.1

Linear mappings of vector spaces

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Problem II.2.1.1

Writing the coordinate column X=[x1,x2,x3,x4]X=\left[x_{1}, x_{2}, x_{3}, x_{4}\right] as a matrix (x1x2x3x4)∈M2(K)\left(\begin{array}{ll}x_{1} & x_{2} \\ x_{3} & x_{4}\end{array}\right) \in M_{2}(\mathfrak {K}), and then taking a fixed matrix A=(a1a2a3a4)∈M2(K)A=\left(\begin{array}{ll}a_{1} & a_{2} \\ a_{3} & a_{4}\end{array}\right) \in M_{2}(\mathfrak {K}), we define two linear transformations:

fL:X↦A(x1x2x3x4)=(x1′x2′x3′x4′)=X′,fR:X↦(x1x2x3x4)A=(x1′′x2′′x3′′x4′′)=X′′, \begin{gathered} f_{L}: X \mapsto A\left(\begin{smallmatrix} x_{1} & x_{2} \\ x_{3} & x_{4} \end{smallmatrix}\right)=\left(\begin{smallmatrix} x_{1}^{\prime } & x_{2}^{\prime } \\ x_{3}^{\prime } & x_{4}^{\prime } \end{smallmatrix}\right)=X^{\prime }, \\ f_{R}: X \mapsto \left(\begin{smallmatrix} x_{1} & x_{2} \\ x_{3} & x_{4} \end{smallmatrix}\right) A=\left(\begin{smallmatrix} x_{1}^{\prime \prime } & x_{2}^{\prime \prime } \\ x_{3}^{\prime \prime } & x_{4}^{\prime \prime } \end{smallmatrix}\right)=X^{\prime \prime }, \end{gathered}

to which correspond the matrices MfLM_{f_{L}} and MfRM_{f_{R}}.

One is asked to verify that

MfL=(a10a200a10a2a30a400a30a4),MfR=(a1a300a2a40000a1a300a2a4) M_{f_{L}}=\left(\begin{smallmatrix} a_{1} & 0 & a_{2} & 0 \\ 0 & a_{1} & 0 & a_{2} \\ a_{3} & 0 & a_{4} & 0 \\ 0 & a_{3} & 0 & a_{4} \end{smallmatrix}\right), \quad M_{f_{R}}=\left(\begin{smallmatrix} a_{1} & a_{3} & 0 & 0 \\ a_{2} & a_{4} & 0 & 0 \\ 0 & 0 & a_{1} & a_{3} \\ 0 & 0 & a_{2} & a_{4} \end{smallmatrix}\right)
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Problem II.2.1.2

Verify the linearity of the following maps:

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(a)

VV is a vector space, W=V/LW=V / L the quotient space, and ff maps every vector x∈V\mathbf{x} \in V to the coset x‾=x+L\overline{\mathbf{x}}=\mathbf{x}+L;

(b)

f:Pn→Pnf: P_{n} \rightarrow P_{n} is the map defined by the rule f(u(t))=tu′(t)−u(t)f(u(t))=t u^{\prime }(t)-u(t).

Find Ker⁡f\operatorname {Ker} f and compute rank⁡f\operatorname {rank} f.

Problem II.2.1.3

Show that the map fC:X↦C−1XCf_{C}: X \mapsto C^{-1} X C, defined by a nonsingular matrix C∈Mn(K)C \in M_{n}(\mathfrak {K}), is linear on Mn(K)M_{n}(\mathfrak {K}) and has the property fC(XY)=fC(X)fC(Y)f_{C}(X Y)=f_{C}(X) f_{C}(Y).

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