Math Problem Statement
Exercise 1 Determine whether the following sets are subspaces of \mathbb{R} ^ 3 under the operations of addition and scalar multiplication defined on \mathbb{R} ^ 3 Justify your answers. W_{1} = {[[x], [y], [z]] \in \mathbb{R} ^ 3, x = 3y, z = - y} 2) W2 = y ER3,2x-7y+z = 0 2. } Z (3) W_{3} = {[[x], [y], [z]] \in \mathbb{R} ^ 3, x ^ 2 - 3z ^ 2 = 0}
Solution
To determine if each set , , and is a subspace of , we need to check three criteria:
- The zero vector is in the set.
- The set is closed under vector addition.
- The set is closed under scalar multiplication.
Let's analyze each set:
1.
Zero vector:
- Check if is in :
- For , and . Thus, .
Closure under addition:
- Take two vectors and : [ [3y_1, y_1, -y_1] + [3y_2, y_2, -y_2] = [3y_1 + 3y_2, y_1 + y_2, -y_1 - y_2] ] Set , then: Hence, the result is in .
Closure under scalar multiplication:
- For scalar and vector : [ c[3y, y, -y] = [3(cy), cy, -cy] ] Set , then: Thus, the result is in .
Conclusion:
- is a subspace of .
2.
Zero vector:
- Check :
Closure under addition:
- Take and : Adding gives: Thus, closure under addition holds.
Closure under scalar multiplication:
- For scalar : Hence, closure under scalar multiplication holds.
Conclusion:
- is a subspace of .
3.
Zero vector:
- Check :
Closure under addition:
- Take and : This implies: However, for : Thus, it may not be in .
Conclusion:
- is not a subspace of .
Summary
- : Subspace
- : Subspace
- : Not a subspace
Do you want further details on any of these points or have any questions?
Here are some follow-up questions:
- How do you prove that a set is not a subspace?
- What are other examples of subspaces in ?
- Can you explain the geometric interpretation of subspaces?
- What conditions must be satisfied for a set to be closed under addition?
- How do you find the dimension of a subspace?
Tip: Always verify the zero vector first when checking for subspaces; it’s a key indicator.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Spaces
Formulas
Closure under addition
Closure under scalar multiplication
Theorems
Subspace Theorem
Suitable Grade Level
Grades 11-12