Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q9E (page 199)

State true or false, if $W_{1}$ and $W_{2}$ are subspaces of a linear space V, then the intersection $W_{1} 鈭� W_{2}$ must be a subspace of V as well.

Short Answer

Expert verified

The given statement is true.

Step by step solution

Step-1 Definition of intersection

Recall that the intersection $U 鈭� V$ is the set of elements that are both elements ofUandV.In the set theoretical notation, we have $U 鈭� V = x x 鈭� U$ And $x 鈭� v$ }.

Step-2 Explanation for the above step

Suppose that vectors, $v_{1}, v_{2}$ are in the intersection of $w_{1} 鈭� w_{2}$ and a, b are scalars. We have to find out if $a v_{1} + b v_{2}$ is in intersection of $w_{1} 鈭� w_{2}$ .

Since, $v_{1}, v_{2}$ are in $w_{1} 鈭� w_{2}$ , then, $v_{1}, v_{2}$ are in both $w_{1}$ and $w_{2}$ . Now, $w_{1}, w_{2}$ are subspaces, so $a v_{1} + b v_{2}$ must be in bothb $w_{1}$ and $w_{2}$ , so it is also in intersection $w_{1} 鈭� w_{2}$ .

This means that the intersection is a subspace.

Hence, the statement is true.

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91影视

State true or false, if $W_{1}$ and $W_{2}$ are subspaces of a linear space V, then the intersection $W_{1} 鈭� W_{2}$ must be a subspace of V as well.

Short Answer

Step by step solution

Step-1 Definition of intersection

Step-2 Explanation for the above step

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91影视

State true or false, if W1andW2are subspaces of a linear space V, then the intersectionW1鈭�W2must be a subspace of V as well.

Short Answer

Step by step solution

Step-1 Definition of intersection

Step-2 Explanation for the above step

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Geometry

Discrete Mathematics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

State true or false, if $W_{1}$ and $W_{2}$ are subspaces of a linear space V, then the intersection $W_{1} 鈭� W_{2}$ must be a subspace of V as well.