https://file.aiquickdraw.com/mathbot/file/1727739787708ohd7p7k1.jpg

In this exercise, we are given a matrix \( A \) and a vector \( b \), with the matrix \( A \) having three column vectors \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \), and \( W = \text{Span} \{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \} \). Let's break down each part of the question.

1. **Is \( \mathbf{b} \in \{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\}? How many vectors are in this set?**
   - The set \( \{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\} \) consists of exactly three vectors. Therefore, the number of vectors in the set is **3**.

2. **Is \( \mathbf{b} \in W \)? How many vectors are in \( W \)?**
   - \( W \) is the span of \( \{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\} \), meaning it consists of all linear combinations of the vectors \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \). The vector \( \mathbf{b} \) can be expressed as a linear combination of \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \) as follows:
   \[
   \mathbf{b} = (-4)\mathbf{a}_1 + (-1)\mathbf{a}_2 + (-2)\mathbf{a}_3
   \]
   Hence, \( \mathbf{b} \in W \), and there are **infinitely many vectors in \( W \)** because the span of three independent vectors is a subspace with infinitely many vectors.

3. **Show that \( \mathbf{a}_2 \) is in \( W \).**
   - The vector \( \mathbf{a}_2 \) is one of the generators of \( W \), meaning it is already part of the set that spans \( W \). Therefore, \( \mathbf{a}_2 \in W \) because \( W = \text{Span} \{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\} \), and any vector in the set of generators automatically belongs to \( W \).

Let me know if you'd like further clarification on any of these points!

### Related Questions:
1. What does it mean for a set of vectors to span a space?
2. How can you determine if a vector is in the span of other vectors?
3. What is the geometric interpretation of the span of vectors?
4. How can we verify linear independence of vectors in a span?
5. How do row operations help in determining whether a vector is in the span of other vectors?

**Tip:** When dealing with subspaces like spans, it's useful to remember that any vector in a subspace can be written as a linear combination of the generators (basis vectors) of that subspace.

Let A = [[1, 0, -2], [0, 3, -4], [-2, 6, 4]] and b = [5, -6]. Denote the columns of A by a_1, a_2, a_3, and let W = Span{a_1, a_2, a_3}. a) Is b in {a_1, a_2, a_3}? How many vectors are in {a_1, a_2, a_3}? b) Is b in W? How many vectors are in W? c) Show that a_2 is in W.

Learn how to determine whether a vector is in the span of other vectors using matrix operations. This detailed problem involves matrix A, vector b, and explores the concepts of linear combinations, span, and vector spaces. Ideal for college-level linear algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation