moltiata C 1. Which of the following sets constitute a vector space? The answer must be in one of the two forms: Jeun V is a vector space because it is closed under addition and multiplication by scalars (this means that mentally you checked that it is closed under these operations and they satisfy the eight required properties) or V is not a vector space because (give a reason and support this reason with an example). bacolo si SMA (a) The set M₁of all 2x2 lower-triangular matrices: {all [ 8 ]}. { a 0 b C (b) The set M₂ of all 6×1, 3×2, 2×3, and 1×6 matrices. (c) The set V₁ of all vectors in the right-hand side of the xy-plane:qa max& a1 V1 strate your answer with V₁ = {all [2₂], ¹20}. 1 Illustrate your answer with a picture. a2 sols drdO (d) The set V₂ all of all vectors on the line y = x in the xy-plane: Illustrate your answer with a picture. If the answer is {all [a]}. negative, also give a second reason (with an example). (e) The set V3 of all vectors of the length one in a plane: where x² M₁ - = (op V₂: = where a₁ > " ibbs wan JadW bau becolo ad posto bmo PO + y² = 1}. Illustrate your answer with a pic- V3 = ture. If the answer is negative, also give a second reason (with an example). (f) The got Or of all fourth-order polynomials with
moltiata C 1. Which of the following sets constitute a vector space? The answer must be in one of the two forms: Jeun V is a vector space because it is closed under addition and multiplication by scalars (this means that mentally you checked that it is closed under these operations and they satisfy the eight required properties) or V is not a vector space because (give a reason and support this reason with an example). bacolo si SMA (a) The set M₁of all 2x2 lower-triangular matrices: {all [ 8 ]}. { a 0 b C (b) The set M₂ of all 6×1, 3×2, 2×3, and 1×6 matrices. (c) The set V₁ of all vectors in the right-hand side of the xy-plane:qa max& a1 V1 strate your answer with V₁ = {all [2₂], ¹20}. 1 Illustrate your answer with a picture. a2 sols drdO (d) The set V₂ all of all vectors on the line y = x in the xy-plane: Illustrate your answer with a picture. If the answer is {all [a]}. negative, also give a second reason (with an example). (e) The set V3 of all vectors of the length one in a plane: where x² M₁ - = (op V₂: = where a₁ > " ibbs wan JadW bau becolo ad posto bmo PO + y² = 1}. Illustrate your answer with a pic- V3 = ture. If the answer is negative, also give a second reason (with an example). (f) The got Or of all fourth-order polynomials with
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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A,B,C,D,E please
![**Problem Statement:**
1. **Which of the following sets constitute a vector space?** The answer must be in one of the two forms:
- **Form 1:** V is a vector space because it is closed under addition and multiplication by scalars (this means that mentally you checked that it is closed under these operations and they satisfy the eight required properties).
- **Form 2:** V is not a vector space because ... (give a reason and support this reason with an example).
**Sets to Consider:**
(a) **The set \( M_1 \) of all 2x2 lower-triangular matrices:**
\[ M_1 = \left\{ \text{all } \begin{bmatrix} a & 0 \\ b & c \end{bmatrix} \right\} \]
(b) **The set \( M_2 \) of all 6x1, 3x2, 2x3, and 1x6 matrices.**
(c) **The set \( V_1 \) of all vectors in the right-hand side of the xy-plane:**
\[ V_1 = \left\{ \text{all } \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}, \text{ where } a_1 \geq 0 \right\} \]
- *Illustrate your answer with a picture.*
(d) **The set \( V_2 \) of all vectors on the line \( y = x \) in the xy-plane:**
\[ V_2 = \left\{ \text{all } \begin{bmatrix} a \\ a \end{bmatrix} \right\} \]
- *Illustrate your answer with a picture. If the answer is negative, also give a second reason (with an example).*
(e) **The set \( V_3 \) of all vectors of the length one in a plane:**
\[ V_3 = \left\{ \text{all } \begin{bmatrix} x \\ y \end{bmatrix}, \text{ where } x^2 + y^2 = 1 \right\} \]
- *Illustrate your answer with a picture. If the answer is negative, also give a second reason (with an example).*
(f) **The set \( Q_6 \) of all fourth-order](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F801d34df-dcab-45aa-85e0-2349cce424a4%2Fe43b4edf-a32c-4eb6-8d97-2057d5a55ece%2Fpw741ze_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
1. **Which of the following sets constitute a vector space?** The answer must be in one of the two forms:
- **Form 1:** V is a vector space because it is closed under addition and multiplication by scalars (this means that mentally you checked that it is closed under these operations and they satisfy the eight required properties).
- **Form 2:** V is not a vector space because ... (give a reason and support this reason with an example).
**Sets to Consider:**
(a) **The set \( M_1 \) of all 2x2 lower-triangular matrices:**
\[ M_1 = \left\{ \text{all } \begin{bmatrix} a & 0 \\ b & c \end{bmatrix} \right\} \]
(b) **The set \( M_2 \) of all 6x1, 3x2, 2x3, and 1x6 matrices.**
(c) **The set \( V_1 \) of all vectors in the right-hand side of the xy-plane:**
\[ V_1 = \left\{ \text{all } \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}, \text{ where } a_1 \geq 0 \right\} \]
- *Illustrate your answer with a picture.*
(d) **The set \( V_2 \) of all vectors on the line \( y = x \) in the xy-plane:**
\[ V_2 = \left\{ \text{all } \begin{bmatrix} a \\ a \end{bmatrix} \right\} \]
- *Illustrate your answer with a picture. If the answer is negative, also give a second reason (with an example).*
(e) **The set \( V_3 \) of all vectors of the length one in a plane:**
\[ V_3 = \left\{ \text{all } \begin{bmatrix} x \\ y \end{bmatrix}, \text{ where } x^2 + y^2 = 1 \right\} \]
- *Illustrate your answer with a picture. If the answer is negative, also give a second reason (with an example).*
(f) **The set \( Q_6 \) of all fourth-order
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