Linear Transformation: Examples and Solutions 1st. Edition by Nita H. Shah, Urmila B. Chaudhari | 9780367613259 | 0367613255 | Holooly

Linear Transformation: Examples and Solutions

37 SOLVED PROBLEMS

Question: 1.4

Find the standard matrix for the linear transformation T defined by the formula T(x1,x2) = (x2,-x1,x1+3×2,x1-x2) . ...

Verified Answer:

The standard matrix for the linear transformation ...

Question: 1.3

Find the standard matrix for the linear transformation defined by the equations ω1 = 2×1-3×2+x4 and ω2 = 3×1+5×2-x3. ...

Verified Answer:

Let linear transformation of the above equation be...

Question: 1.5

Find the standard matrix for the linear operator T defined by the formula T(x1,x2,x3)=(4×1,7×2,-8×3). ...

Verified Answer:

The standard matrix for the linear operator [latex...

Question: 1.6

Let T1 : R² → R² and T2 : R² → R³ be the transformations given by T1(x,y) = (x+y,y) and T2 (x,y) = (2x,y,x+y). Find the formula for T2∘T1 (x,y). ...

Verified Answer:

Let the standard matrix for

T_{1}\left(x,y\...

Question: 1.7

Find the standard matrix for the linear operator T : R² → R² that is a dilation of a vector with factor k = 2, then the resulting vector is rotated by θ = 45°, and then contraction is applied with factor k = √2. ...

Verified Answer:

The linear operator T can be expressed as the comp...

Question: 3.1

Determine whether the linear transformation T:R² → R², where T(x,y)=(x+y,x-y), is one-one, onto, or both or neither. ...

Verified Answer:

A linear transformation is one-one if and only if ...

Question: 4.7

Let T:R² → R² be defined by T([x1 x2])=[x1+x2 -2×1+4×2]. Find the matrix for T with respect to the basis S1={e1,e2} for R², and also find a matrix for T with respect to the basis S2={u1′,u2′} for R², where u1′=[1 1] and u2′=[1 2]. ...

Verified Answer:

T(e_{1})=T\left\lgroup{\left[\begin{array}...

Question: 4.6

Show that the matrices [1 1 -1 4] and [2 1 1 3] are similar but that [3 1 -6 -2] and [-1 2 1 0] are not. ...

Verified Answer:

Let

A_{1}={\left[\begin{array}{c c}{1}&...

Question: 4.4

Let T:R² → R² be the linear operator defined by T([x1 x2])=[x1+x2 -2×1+4×2]. Find the matrix of the operator T with respect to the basis S={u1,u2}, where u1=[1 1] and u2=[1 2]. Also, verify [T]S[X]S=[T(X)]S. ...

Verified Answer:

Here,

T(u_{1})=T\left\lgroup\left[\ \begin{...

Question: 3.9

Let T:R² → R² and S:R² → R² be the linear operators given by the formula T(x,y)=(x+y,x-y) and S(x,y)=(2x+y,x-2y). i. Show that T and S are one-one. ii. Find formulas for T^−1(x,y) and S^−1(x,y) and (T∘S)^−1(x,y). iii. Verify (T∘S)^−1=S^−1∘T^−1. ...

Verified Answer:

i. T and S are one-one if ker(T)={0} and ker(S)={0...

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