Other
MAT2611 Sem ONE Assign FOUR 2022
- Course
- Institution
UNISA Linear Algebra MAT2611 Semester ONE Assignment FOUR of 2022 solutions. Vector spaces Subspaces Span Basis Dimension Linear independence
[Show more]
Seller
Follow
lyzo2005
Reviews received
Content preview
MAT2611 SEMESTER 1 ASSIGNMENT 4 2022Problem 11
𝑇ℎ𝑒 𝑠𝑒𝑡 {(3, 1, 4), (2, −3, 5), (5, 9, 𝑡)} 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑛𝑜𝑡 𝑙𝑖𝑛𝑒𝑎𝑟𝑙𝑦 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡.
𝑀𝑒𝑡ℎ𝑜𝑑 1
𝑂𝑛 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑘1 (3, 1, 4) + 𝑘2 (2, −3, 5) + 𝑘3 (5, 9, 𝑡) = (0, 0, 0)
𝑇ℎ𝑒 𝑘𝑖 𝑤𝑖𝑙𝑙 𝑛𝑜𝑡 𝑏𝑒 𝑎𝑙𝑙 𝑒𝑞𝑢𝑎𝑙 𝑧𝑒𝑟𝑜
(𝑘1 , 𝑘2 , 𝑘3 ) ≠ (0, 0, 0)
(3𝑘1 , 1𝑘1 , 4𝑘1 ) + (2𝑘2 , −3𝑘2 , 5𝑘2 ) + (5𝑘3 , 9𝑘3 , 𝑡𝑘3 ) = (0, 0, 0)
(3𝑘1 + 2𝑘2 + 5𝑘3 , 1𝑘1 − 3𝑘2 + 9𝑘3 , 4𝑘1 + 5𝑘2 + 𝑡𝑘3 ) = (0, 0, 0)
3𝑘1 + 2𝑘2 + 5𝑘3 = 0
1𝑘1 − 3𝑘2 + 9𝑘3 = 0
4𝑘1 + 5𝑘2 + 𝑡𝑘3 = 0
𝑇ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 𝑠𝑦𝑠𝑡𝑒𝑚 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 𝑚𝑢𝑠𝑡 𝑛𝑜𝑡 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑢𝑛𝑖𝑞𝑢𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (0, 0, 0).
𝐼𝑡 𝑚𝑢𝑠𝑡 ℎ𝑎𝑣𝑒 𝑎𝑛 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠.
3 2 5 0
[1 −3 9| 0]
4 5 𝑡 0
1
𝑅2 → 𝑅2 − 𝑅1
3
4
𝑅3 → 𝑅3 − 𝑅1
3
3 2 5 0
1 1 1 1
1 − (3) −3 − (2) 9 − (5)| 0 − (0)
3 3 3 | 3
4 4 4 4
4 − (3) 5 − (2) 𝑡 − (5) 0 − (0)
[ 3 3 3 3 ]
3 2 5
11 22 0
0 − |
3 3 |0
7 20 0
[0 3
𝑡−
3 ]
, 3 2 5
11 22 0
0 − |
3 3 |0
7 20 0
[0 3
𝑡−
3 ]
𝑅2 → 3𝑅2
𝑅3 → 3𝑅3
3 2 5
11 22 0
3(0) 3 (− ) 3( ) | ( )
3 3 3 0
|
7 20 3(0)
[3(0) 3( )
3
3 (𝑡 − )
3 ]
3 2 5 0
[0 −11 22 | 0]
0 7 3𝑡 − 20 0
7
𝑅3 → 𝑅3 + 𝑅
11 2
3 2 5 0
0 −11 22 0
[ 7 7 | 7 ]
0 7 + (−11) 3𝑡 − 20 + (22) 0 + (0)
11 11 11
3 2 5 0
[0 −11 22 | 0]
0 0 3𝑡 − 6 0
𝑇ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑤𝑖𝑙𝑙 ℎ𝑎𝑣𝑒 𝑎𝑛 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 𝑖𝑓:
3𝑡 − 6 = 0
3𝑡 = 6
6
𝑡=
3
𝑡=2
𝑇ℎ𝑒 𝑠𝑒𝑡 {(3, 1, 4), (2, −3, 5), (5, 9, 2)} 𝑖𝑠 𝑛𝑜𝑡 𝑙𝑖𝑛𝑒𝑎𝑟𝑙𝑦 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡.
𝑀𝑒𝑡ℎ𝑜𝑑 2
𝑇ℎ𝑒 𝑠𝑒𝑡 {(3, 1, 4), (2, −3, 5), (5, 9, 𝑡)} 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑛𝑜𝑡 𝑙𝑖𝑛𝑒𝑎𝑟𝑙𝑦 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡.
𝑇ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 (5, 9, 𝑡) 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 (3, 1, 4) 𝑎𝑛𝑑 (2, −3, 5)
(5, 9, 𝑡) = 𝑎1 (3, 1, 4) + 𝑎2 (2, −3, 5) 𝑤ℎ𝑒𝑟𝑒 𝑎1 𝑎𝑛𝑑 𝑎2 ∈ ℝ
(5, 9, 𝑡) = (3𝑎1 , 1𝑎1 , 4𝑎1 ) + (2𝑎2 , −3𝑎2 , 5𝑎2 )
(5, 9, 𝑡) = (3𝑎1 + 2𝑎2 , 1𝑎1 − 3𝑎2 , 4𝑎1 + 5𝑎2 )
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller lyzo2005. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.92. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
74735 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now