Q66E If the matrix of a linear trans... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q66E (page 201)

If the matrix of a linear transformation T with respect to basis isthen there must exist a non-zero element f in the domain of T such that T(f)=5f.

Short Answer

Expert verified

The givens statement is false.

Step by step solution

Find the standard bases from the given matrix.

Considerthe matrix $[\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$ of size $2 脳 2$ of a linear transformation T.

Let $e_{1}$ and $e_{2}$ be the standard bases of domain and co-domain of T such that

$T (e_{1}) = 3 e_{1} T (e_{2}) = 5 e_{1} + 4 e_{2}$

Proof by contradiction.

Assume f to be a non-zero element of domain of T such that $T (f) = 5 f$

Then consider the input $f (x)$ and the output $T (f (x))$ in coordinates with respect to the standard bases as follows,

$\begin{array}{rcl} f & = & a e_{1} + b e_{2} \\ T (f) & = & T (a e_{1} + b e_{2}) \\ 5 f & = & a T (e_{1}) + b T (e_{2}) \\ 5 f & = & a (3 e_{1}) + b (5 e_{1} + 4 e_{2}) \end{array}$

Simplify further to find the value of a and b.

$\begin{array}{rcl} 5 (a e_{1} + b e_{2}) & = & a (3 e_{1}) + b (5 e_{1} + 4 e_{2}) \\ 5 a e_{1} + 5 b e_{2} & = & 3 a e_{1} + 5 b e_{1} + 4 b e_{2} \\ b & = & 4 b, 鈥� 2 a = 5 b \end{array}$

Then,

$b = 0 a = 0$

Therefore,

$f = a e_{1} + b e_{2} f = 0$

Thus, f is a zero element which is a contradiction to our assumption.

Hence, there exists a non-zero element f in the domain of T such that $T (f) = 5 f$ is not possible.

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91影视

If the matrix of a linear transformation T with respect to basis isthen there must exist a non-zero element f in the domain of T such that T(f)=5f.

Short Answer

Step by step solution

Find the standard bases from the given matrix.

Proof by contradiction.

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