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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q8E (page 184)

Question: Find the transformation is linear and determine whether they are isomorphism.

Short Answer

Expert verified

The solution is not linear.

Step by step solution

01

Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

$T (f + g) = T (f) + T (g) T (kf) = kT (f)$

For all elements f,g of V and k is scalar.

An invertible linear transformation is called an isomorphism.

02

Explanation of the solution

The given transformation as follows.

$T (M) = M [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$ from $R^{2 脳 2} to R^{2 脳 2}$ .

By using the definition of linear transformation as follows.

$T (A + B) = T (A) + T (B) T (kA) = kT (A)$

Consider the $2 脳 2$ matrix for $l_{2}$ as follows.

$l_{2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Substitute the value 2 for k in $T (2 l_{2})$ and simplify as follows.

$T (2 l_{2}) = (2 l_{2}) [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] (2 l_{2}) = [\begin{matrix} 2 & 4 \\ 6 & 8 \end{matrix}] (2 l_{2}) T (2 l_{2}) = [\begin{matrix} 4 & 8 \\ 12 & 16 \end{matrix}]$

Similarly, simplify for as follows.

$2 T (l_{2}) = 2 (l_{2}) [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] (l_{2}) 2 T (l_{2}) = [\begin{matrix} 2 & 4 \\ 6 & 8 \end{matrix}]$

Since, $T (2 l_{2}) 鈮� 2 T (l_{2}) and T$ is not linear.

Hence, the transformation T is not linear.

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Most popular questions from this chapter

Find the basis of all $3 脳 3$ upper triangular matrix, and determine its dimension.

(a) Show that T is a linear transformation.

(b) Find the kernel of T.

(c) Show that the image of T is a space $L (R^{m}, R^{n})$ of all linear transformation $R^{m}$ to role="math" localid="1659420398933" $R^{n}$ .

(d) Find the dimension of $L (R^{m}, R^{n})$ .

Find the image, kernel, rank, and nullity of the transformation $T$ in $T (f (t)) = [\begin{matrix} f (7) \\ f (11) \end{matrix}]$ from $P_{2}$ to $R$ .

Define an isomorphism from $P_{3}$ to $R^{2 脳 2}$ .

TRUE OR FALSE?

10. If T is a linear transformation from $P_{6} to R^{2 脳 2}$ , then the kernel of T must be three-dimensional.

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