Math Problem Statement

Seien V = R^4, U = ⟨(1, -1, 0, 0), (1, 0, -1, 0)⟩ und W = ⟨(1, -1, 0, 0), (1, -1, -1, 0), (0, 0, 1, 0)⟩. Bestimmen Sie Basen von U + W und U ∩ W.

Solution

The problem is asking us to determine bases for U+WU + W and UWU \cap W, where:

  • V=R4V = \mathbb{R}^4
  • UU and WW are subspaces of VV, spanned by specific vectors.

The subspaces are defined as follows:

  • U=span{(1100),(1010)}U = \text{span} \left\{ \begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \end{pmatrix} \right\}
  • W=span{(1100),(1110),(0010)}W = \text{span} \left\{ \begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ -1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \right\}

To find bases for U+WU + W and UWU \cap W, we proceed with the following steps:

Step 1: Find a Basis for UU

The basis vectors for UU are: (1100)and(1010)\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \end{pmatrix} We’ll verify if these two vectors are linearly independent. In this case, they are independent, so they form a basis for UU.

Step 2: Find a Basis for WW

The basis vectors for WW are: (1100),(1110),(0010)\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ -1 \\ -1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} We should check if these vectors are linearly independent. It turns out that they are linearly independent, so these three vectors form a basis for WW.

Step 3: Find a Basis for U+WU + W

To find a basis for U+WU + W, we combine all basis vectors of UU and WW: (1100),(1010),(1110),(0010)\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ -1 \\ -1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} We then determine the linearly independent vectors among them. After applying Gaussian elimination, we find that a basis for U+WU + W consists of: (1100),(1010),(0010)\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}

Step 4: Find a Basis for UWU \cap W

To find a basis for UWU \cap W, we need to find all vectors that can be expressed as a linear combination of both bases of UU and WW. This is equivalent to solving a system that satisfies both sets of linear equations represented by the basis vectors of UU and WW. Solving this system reveals that the only common vector is: (1100)\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix} Thus, a basis for UWU \cap W is: (1100)\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix}

Summary

  • Basis for U+WU + W: {(1100),(1010),(0010)}\left\{ \begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \right\}
  • Basis for UWU \cap W: {(1100)}\left\{ \begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix} \right\}

Would you like further details or have any questions?

Related Questions

  1. How do you determine if a set of vectors is linearly independent?
  2. What are the steps to find the span of a set of vectors?
  3. How can we use Gaussian elimination to find bases for vector spaces?
  4. What are some applications of finding the intersection and sum of subspaces?
  5. How does the rank-nullity theorem relate to subspace intersections?

Tip

Remember that UWU \cap W always consists of vectors that satisfy both subspace conditions, while U+WU + W combines all vectors in either UU or WW, filtering out redundancies.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Basis of Subspaces
Intersection of Subspaces
Span of Vectors

Formulas

Gaussian elimination for determining linear independence
Definition of vector span
Basis determination through elimination

Theorems

Subspace Sum Theorem
Intersection of Subspaces
Linear Independence

Suitable Grade Level

Undergraduate (University level)