Math Problem Statement

A set of 5 vectors in R^4, analyzing conditions for basis and span, and determining the span of a matrix in R^5.

Solution

Let's go through each of these questions one by one.

Problem 2

Question: A set of 5 vectors in R4\mathbb{R}^4.

Option Analysis:

  1. (a) Can be a basis if the set is linearly independent:

    • To form a basis in R4\mathbb{R}^4, we only need 4 linearly independent vectors. Since there are 5 vectors here, they cannot all be linearly independent in R4\mathbb{R}^4; otherwise, they would span a space with more than 4 dimensions, which is impossible in R4\mathbb{R}^4.
    • Therefore, this statement is false.

  2. (b) Cannot span R4\mathbb{R}^4:

    • In R4\mathbb{R}^4, any set of more than 4 vectors is linearly dependent. Although this set of 5 vectors cannot be a basis due to dependence, it can still span R4\mathbb{R}^4 if 4 of them are linearly independent.
    • This statement is also false.

  3. (c) Spans R4\mathbb{R}^4 if 4 vectors in the set are linearly independent:

    • If 4 of the vectors are linearly independent, they would span R4\mathbb{R}^4. The additional vector (even if it is dependent) does not impact the span.
    • This statement is true.

Answer for Problem 2: (c)

Problem 3

Question: Find the span of the given matrix in terms of its subspace in R5\mathbb{R}^5.

The matrix in question has the following columns:

\begin{bmatrix} 0 \\ -2 \\ 2 \\ -3 \\ -3 \end{bmatrix}, \begin{bmatrix} -2 \\ -4 \\ 2 \\ 3 \\ 6 \end{bmatrix}$$ **Determining the Dimensionality of the Span**: 1. These vectors are in $$\mathbb{R}^5$$, meaning each vector has 5 components. 2. To determine the span’s dimensionality, we need to see if these vectors are linearly independent or dependent. By inspecting the vectors: - We can see that the third vector, $$\begin{bmatrix} -2 \\ -4 \\ 2 \\ 3 \\ 6 \end{bmatrix}$$, is a linear combination of the first two vectors. Specifically: $$-2 \times \begin{bmatrix} 1 \\ -1 \\ 1 \\ 2 \\ 0 \end{bmatrix} + 1 \times \begin{bmatrix} 0 \\ -2 \\ 2 \\ -3 \\ -3 \end{bmatrix} = \begin{bmatrix} -2 \\ -4 \\ 2 \\ 3 \\ 6 \end{bmatrix}$$ - This shows that the three vectors are linearly dependent and thus span a subspace of dimension 2. **Option Analysis**: 1. **(a) Is a 4-dimensional subspace in $$\mathbb{R}^5$$** - Incorrect because the span is only 2-dimensional. 2. **(b) Is a 3-dimensional subspace in $$\mathbb{R}^5$$** - Incorrect because the span is only 2-dimensional. 3. **(c) Is a 2-dimensional subspace in $$\mathbb{R}^5$$** - Correct, as shown above. **Answer for Problem 3**: **(c)** --- Would you like a deeper explanation of any of these solutions? ### Further Related Questions 1. How do we determine if a set of vectors is linearly dependent or independent? 2. Why does the dimension of a basis for $$\mathbb{R}^n$$ equal $$n$$? 3. Can a set of vectors span $$\mathbb{R}^n$$ if it contains more than $$n$$ vectors? Why or why not? 4. How can we find a basis for the span of a set of vectors? 5. What is the significance of the rank of a matrix in determining its span? **Tip:** When checking for linear dependence among vectors, look for possible linear combinations that can express one vector as a sum or multiple of others.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Independence
Span

Formulas

-

Theorems

Basis of a Vector Space
Linear Independence and Dependence
Span Theorem

Suitable Grade Level

Undergraduate

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