Math Problem Statement

Find a list of four distinct vectors in F^3 whose span equals {(x, y, z) ∈ F^3 : x + y + z = 0}.

Solution

I see that you’ve uploaded an image from the book Linear Algebra Done Right by Sheldon Axler, specifically from the section on span and linear independence (Exercises 2A). Here are some of the questions presented in the image:

  1. Find a list of four distinct vectors in F3\mathbb{F}^3 whose span equals {(x,y,z)F3:x+y+z=0}\{(x, y, z) \in \mathbb{F}^3 : x + y + z = 0 \}.

  2. Prove or give a counterexample: If v1,v2,v3,v4v_1, v_2, v_3, v_4 spans VV, then the list v1v2,v2v3,v3v4,v4v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4 also spans VV.

  3. Suppose v1,,vmv_1, \dots, v_m is a list of vectors in VV. For k{1,,m}k \in \{1, \dots, m \}, let wk=v1++vkw_k = v_1 + \dots + v_k. Show that span(v1,,vm)=span(w1,,wm)\text{span}(v_1, \dots, v_m) = \text{span}(w_1, \dots, w_m).

  4. (a) Show that a list of length one in a vector space is linearly independent if and only if the vector in the list is not 0. (b) Show that a list of length two in a vector space is linearly independent if and only if neither of the two vectors in the list is a scalar multiple of the other.

  5. Find a number tt such that (3,1,4),(2,3,5),(5,9,t)(3, 1, 4), (2, -3, 5), (5, 9, t) is not linearly independent in R3\mathbb{R}^3.


Would you like to focus on a particular problem from this list or go over a solution in detail?

Here are some related questions to expand the concepts:

  1. What are the conditions for a set of vectors to span a vector space?
  2. How do you check if a set of vectors is linearly independent?
  3. What is the geometric interpretation of spanning in R3\mathbb{R}^3?
  4. How does changing a single vector in a set affect the linear independence of the set?
  5. Can you provide an example of a set of linearly dependent vectors in R3\mathbb{R}^3?

Tip: To prove linear independence, remember that a set of vectors is linearly independent if and only if the only solution to the equation c1v1+c2v2++cnvn=0c_1 v_1 + c_2 v_2 + \dots + c_n v_n = 0 is c1=c2==cn=0c_1 = c_2 = \dots = c_n = 0.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Span
Linear Independence

Formulas

Span Definition: span(v1, v2, ..., vn) is the set of all linear combinations of the vectors
Linear Independence: A set of vectors is linearly independent if no vector in the set is a linear combination of the others.

Theorems

Basis Theorem
Linear Independence Definition
Span Theorem

Suitable Grade Level

Undergraduate level (Math or Engineering students)