# Question 2.1: In R2, fix the following points O = (1, 0), A1 = (1, 2), A2 ...

In R², fix the following points

O = (1, 0), $A_{1}$ = (1, 2), $A_{2}$ = (0, 1), and

$O^{′}=$ (−1,−1), $B_{1}$ = (0, 0), $B_{2}$ = (2, 3).

Construct the vectorized spaces Σ(O ; $A_{1}$ , $A_{2}$) and Σ($O^{′}$; $B_{1}$ , $B_{2}$), and then use them to justify the content of (2.4.2).

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Question: 2.9

## The linear operator A=[0 1 b a], where ab ≠ 0 has the following properties:(1) a2 + 4b > 0. Let λ1 =a+√a2+4b/2 and λ2 = a-√a2+4b/2 and vi^→ = (b, λi) for i = 1, 2.1. vi^→A = λi v i^→, i = 1, 2. Thus, λ1 and λ2 are eigenvalues of A with corresponding eigenvectors v1^→ and v2^→.2. <> and< ...

Is there any more basic operator than these mentio...
Question: 2.25

## Study the affine transformation T(x^→ ) = (−2, 2√3) + x^→A, where A =[1/2 √3/2 √3/2 -1/2 ].   ...

Although A is orthogonal, i.e. A^{*}=A^{-1}...
Question: 2.24

## In N = {e1^→, e2^→}, let f: R2 → R2 be defined as f(x1, x2) =(x1 +1/2 x2,−1/2 x1 + 2×2) = x^→A, where x^→ = (x1, x2) and A =[1 -1/2 1/2 2 ] . Investigate mapping properties of A. ...

$f$ is isomorphic. Its characteristi...
Question: 2.16

## Using N = {e1^→,e2^→}, the Cartesian coordinate system, let the linear operator f: R2 → R2 be defined as f(x1, x2) = (2×1 − 3×2,−4×1 + 6×2) = x^→A, where x^→ = (x1, x2) and A =[2 -4 -3 6 ]. ...

$f$ is obviously linear. Now, the ker...
Question: 2.14

## Let R2 be endowed with Cartesian coordinate system N = {e1^→,e2^→} as Fig. 2.17(b) indicated. Investigate the geometric mapping properties of the linear transformation f(x1, x2) = (2×1 − x2,−3×1 + 4×2) = x^→A, where x^→ = (x1, x2) and A= [2 -3 -1 4 ]. ...

It is easy to check that $f$ is inde...
Question: 2.12

## Let A = [1 2 2 -7]. (1) Solve the equation Ax ∗^→ = b ∗^→ where x ^→= (x1, x2) and  b^→ = (b1, b2). (2) Investigate the geometric mapping properties of A. ...

As against \overrightarrow{x}A=\overrightar...
Question: 2.11

## Let A =[0 2 -1 1 ]. Do the same problems as in Example 1. ...

For  \overrightarrow{x}=\left(x_{1},x_{2} \...
Question: 2.13

## Let A =[2 3 -4 -6 ].(1) Solve the equation Ax∗^→ = b∗^→ .(2) Try to investigate the geometric mapping properties of A. ...

Write  A\overrightarrow{x^{*} } =\overright...
Question: 2.10

## Let A= [1 2 4 10] . (1) Solve the equation x^→A =  b^→ where x^→ = (x1, x2) ∈ R2 and  b^→ = (b1, b2) is a constant vector.(2) Investigate the geometric mapping properties of A. ...

Where written out, \overrightarrow{x}A=\ove...