Consider the vectors 10 11 -3 -22 1 -20 9. -32 (a) These vectors are linearly dependent. Write one of them as a linear combination of the others. (b) Find a linearly independent subset of this set of vectors that has the same span (in other words, find a basis for the subspace spanned by these vectors).
Consider the vectors 10 11 -3 -22 1 -20 9. -32 (a) These vectors are linearly dependent. Write one of them as a linear combination of the others. (b) Find a linearly independent subset of this set of vectors that has the same span (in other words, find a basis for the subspace spanned by these vectors).
Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Please explain part a and b. ( I have attached an answer key).
![Consider the vectors
10
11
8
-20
32
(a) These vectors are linearly dependent. Write one of them as a linear combination of the others.
(b) Find a linearly independent subset of this set of vectors that has the same span (in other words, find a
basis for the subspace spanned by these vectors).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2Fefe84ed2-d666-4ced-97b6-5bdbcba62276%2Fjxhsm2ij_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the vectors
10
11
8
-20
32
(a) These vectors are linearly dependent. Write one of them as a linear combination of the others.
(b) Find a linearly independent subset of this set of vectors that has the same span (in other words, find a
basis for the subspace spanned by these vectors).
![1
-2
10
11
-3
4
8
, you will find that its
-22
For part (a), if you row-reduce the coefficient matrix
1
-20
2
-32
1 0
2
1
-4 0
It follows from examining this RREF that one possible answer is
1
RREF is
0
10
1
-22
-3
- 4
4
= 2
-32
although other correct answers are possible. For part (b), since the first, second, and fourth columns of the
coefficient matrix have pivots in the RREF, a basis for the span of all four columns consists of these three
only:
11
8
9.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2Fefe84ed2-d666-4ced-97b6-5bdbcba62276%2Fo8ovw9q_processed.png&w=3840&q=75)
Transcribed Image Text:1
-2
10
11
-3
4
8
, you will find that its
-22
For part (a), if you row-reduce the coefficient matrix
1
-20
2
-32
1 0
2
1
-4 0
It follows from examining this RREF that one possible answer is
1
RREF is
0
10
1
-22
-3
- 4
4
= 2
-32
although other correct answers are possible. For part (b), since the first, second, and fourth columns of the
coefficient matrix have pivots in the RREF, a basis for the span of all four columns consists of these three
only:
11
8
9.
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