Question 6: Let V be a vector space over R, and let T: V→ R4 be a linear map. Suppose 01, 02, 03, 04 € V are such that T(₁) = (1,0,0,0), T(7₂) = (1,1,0,0), T(73) = (1,1,1,0), T(4) = (1,1,1,1). Prove that dim(V) > 4.
Question 6: Let V be a vector space over R, and let T: V→ R4 be a linear map. Suppose 01, 02, 03, 04 € V are such that T(₁) = (1,0,0,0), T(7₂) = (1,1,0,0), T(73) = (1,1,1,0), T(4) = (1,1,1,1). Prove that dim(V) > 4.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Linear Map and Vector Space Dimensions
**Question 6:**
Let \( V \) be a vector space over \( \mathbb{R} \), and let \( T : V \to \mathbb{R}^4 \) be a linear map. Suppose \( \vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4 \in V \) are such that
\[ T(\vec{v}_1) = (1, 0, 0, 0), \]
\[ T(\vec{v}_2) = (1, 1, 0, 0), \]
\[ T(\vec{v}_3) = (1, 1, 1, 0), \]
\[ T(\vec{v}_4) = (1, 1, 1, 1). \]
Prove that \( \dim(V) \geq 4 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cf4b148-a9ff-447b-bf88-730c6d11afaf%2F7b472979-2365-4e83-aa30-07674fdcfc2a%2Fcx1pgzm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Map and Vector Space Dimensions
**Question 6:**
Let \( V \) be a vector space over \( \mathbb{R} \), and let \( T : V \to \mathbb{R}^4 \) be a linear map. Suppose \( \vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4 \in V \) are such that
\[ T(\vec{v}_1) = (1, 0, 0, 0), \]
\[ T(\vec{v}_2) = (1, 1, 0, 0), \]
\[ T(\vec{v}_3) = (1, 1, 1, 0), \]
\[ T(\vec{v}_4) = (1, 1, 1, 1). \]
Prove that \( \dim(V) \geq 4 \).
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