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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: Q6E (page 176)

Which of the subsets Vof ${鈩�}^{3 x 3}$ given in Exercise 6through 11are subspaces of ${鈩�}^{3 x 3}$ . The invertiblematrices.

Short Answer

Expert verified

The invertible 3x3 matrices subset V of ${鈩�}^{3 x 3}$ is not a subspace of ${鈩�}^{3 x 3}$ .

Step by step solution

01

Definition of subspace.

A subset Wof a linear spaceis called a subspace ofif

(a) W contains the neutral element 0of V.

(b) W is closed under addition (if fand gare in Wthen so is f+g)

(c) W is closed under scalar multiplication (if fis in Wand kis scalar, then kfis in W).

we can summarize parts b and c by saying that W is closed under linear combinations.

02

Verification whether the subset is closed under addition and scalar multiplication.

Consider two 3x3 matrix namely,

$A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] B = [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] |A| = 1 (1 - 0) - 0 + 0 |A| = 1$

Similarly, we can write as

$|B| = - 1 (1 - 0) - 0 + 0 |B| = - 1$

Therefore,A,B are invertible.

$A + B = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

The sum of A,B is zero matrix which is not invertible. Hence, invertible matrices are not a subspace of ${鈩�}^{3 x 3}$ .

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Most popular questions from this chapter

In Exercises 5 through 40, find the matrix of the given linear transformation $T$ with respect to the given basis. If no basis is specified, use standard basis: $渭 = (1, t, t)$ for $P_{2}$ ,

$渭 = ([\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}])$

for ${鈩�}^{2 脳 2}$ and $渭 = (1, i)$ for, ${鈩�}^{2 脳 2}$ .For the space $U^{2 脳 2}$ of upper triangular $2 脳 2$ matrices, use the basis

$渭 = ([\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}])$

Unless another basis is given. In each case, determine whether $T$ is an isomorphism. If $T$ isn鈥檛 an isomorphism, find bases of the kernel and image of $T$ and thus determine the rank of $T$ .

17. $T (z) = iz$ from $C$ to $C$ with respect to the basis $B = {1, i}$ .

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (f (t)) = {鈭�}_{- 2}^{3} f (t) d t$ from to $P_{2} t o 鈩�$ .

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

Find the basis of all 2X2diagonal matrix,and determine its dimension.

Find the set of all polynomial $f (t)$ in $P_{2}$ such that $f (1) = 0$ , and determine its dimension.

See all solutions

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