Math Problem Statement

(1 point) Are the following statements true or false?

False

  1. If 𝑆1 and 𝑆2 are subspaces of 𝑅𝑛 of the same dimension, then 𝑆1=𝑆2 .

True 2. If 𝑆= span{𝑒1,𝑒2,𝑒3 }, then π‘‘π‘–π‘š(𝑆)=3 .

True 3. If the set of vectors π‘ˆ is linearly independent in a subspace 𝑆 then vectors can be added to π‘ˆ to create a basis for 𝑆

False 4. If the set of vectors π‘ˆ spans a subspace 𝑆 , then vectors can be added to π‘ˆ to create a basis for 𝑆

True 5. If the set of vectors π‘ˆ spans a subspace 𝑆 , then vectors can be removed from π‘ˆ to create a basis for 𝑆

Solution

Let's review and evaluate the given statements individually:

  1. False:
    This statement is indeed false. Two subspaces S1S_1 and S2S_2 of Rn\mathbb{R}^n can have the same dimension but still be different. For example, different lines through the origin in R2\mathbb{R}^2 can both have dimension 1 but are distinct subspaces.

  2. True:
    This statement is true. If S=span{u1,u2,u3}S = \text{span}\{u_1, u_2, u_3\} and the vectors u1,u2,u3u_1, u_2, u_3 are linearly independent, then the dimension of the span (subspace SS) is equal to the number of independent vectors, which is 3.

  3. True:
    This is true. If a set of vectors UU is linearly independent in a subspace SS, you can always add more vectors (if necessary) to UU to form a basis for SS, thereby completing the spanning set.

  4. False:
    This statement is false. If the set of vectors UU already spans the subspace SS, then adding more vectors to UU will not create a basis. In fact, it will introduce redundancy and linear dependence. A basis for SS requires only the minimal set of linearly independent vectors that span SS.

  5. True:
    This statement is true. If a set of vectors UU spans a subspace SS, you can remove redundant vectors (i.e., linearly dependent vectors) from UU to create a basis for SS, which is the smallest set of linearly independent vectors that span the space.

Would you like further clarification on any of these explanations?

Here are 5 related questions to expand your understanding:

  1. How do you determine if a set of vectors is linearly independent?
  2. What is the difference between a spanning set and a basis?
  3. Can two subspaces of different dimensions in Rn\mathbb{R}^n be equal?
  4. Why is linear dependence important when constructing a basis?
  5. How do you find the dimension of a subspace given a set of vectors?

Tip: When building a basis, focus on identifying the linearly independent vectorsβ€”this ensures that you span the space without unnecessary redundancy.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Subspaces
Spanning Sets
Basis
Linear Independence

Formulas

Dimension of a subspace: dim(S) = number of linearly independent vectors
Span: S = span{v1, v2, ..., vn}

Theorems

Subspace Theorem
Linear Independence Theorem
Basis and Spanning Set Theorem

Suitable Grade Level

University Level (Introductory Linear Algebra)