Short Answer

TRUE OR FALSE?
38. There exists a subspace of $ℝ^{3 \times 4}$ that is isomorphic to $P_{9}$ .

The given statement is true.

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Step by Step Solution

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TABLE OF CONTENTS

Step 1: Isomorphism of vector space

Two vector spaces V and W over the same field F are isomorphic if there is a bisection $T : V \to W$ which preserves addition and scalar multiplication, that is, for all vectors u and v in V, and all scalars $c \in F, T (u + v) = T (u) + T (v) and T (cv) = cT (v)$ .The correspondence T is called an isomorphism of vector spaces.

When $T : V \to W$ is an isomorphism, $T : V' \to W$ then it’s emphasize that it is an isomorphism When V and W, are isomorphic, but the specific isomorphism is not named, we’ll just write $V ~ = W .$ .

Step 2: condition of a subspace to be isomorphic.

Space $ℝ^{3 \times 4}$ is 12-dimensional and $P_{9}$ is 10-dimensional. Spaces of same dimension are isomorphic, so any 10-dimensional subspace of $ℝ^{3 \times 4}$ is isomorphic to $P_{9}$ .

Hence, the statement is true.

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Linear Algebra With Applications

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Short Answer

TRUE OR FALSE?
38. There exists a subspace of $ℝ^{3 \times 4}$ that is isomorphic to $P_{9}$ .

Step by Step Solution

Step 1: Isomorphism of vector space

Step 2: condition of a subspace to be isomorphic.

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Linear Algebra With Applications

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Short Answer

TRUE OR FALSE?38. There exists a subspace of ℝ3×4that is isomorphic toP9.

Step by Step Solution

Step 1: Isomorphism of vector space

Step 2: condition of a subspace to be isomorphic.

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TRUE OR FALSE?
38. There exists a subspace of $ℝ^{3 \times 4}$ that is isomorphic to $P_{9}$ .