4) Consider the vector space M22 of all 2 x 2 matrices with normal matrix addition and scalar multiplication as the operations. Answer the following questions: 4.1) Let W be the set of all matrices A € M22 such that Prove or disprove that W is a subspace of M22. (4[A1])"=^ [A 1² AT
4) Consider the vector space M22 of all 2 x 2 matrices with normal matrix addition and scalar multiplication as the operations. Answer the following questions: 4.1) Let W be the set of all matrices A € M22 such that Prove or disprove that W is a subspace of M22. (4[A1])"=^ [A 1² AT
Elementary Linear Algebra (MindTap Course List)
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Chapter4: Vector Spaces
Section4.6: Rank Of A Matrix And Systems Of Linear Equations
Problem 67E: Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many...
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![4) Consider the vector space M22 of all 2 x 2 matrices with normal matrix addition
and scalar multiplication as the operations. Answer the following questions:
T
T
1
4.1) Let W be the set of all matrices A E M22 such that ( A
= AT
Prove or disprove that W is a subspace of M2.
A1 =
[2 A]
A2
3 0
[o 2]
A3
4.2)
Let
and A4
be matrices in
M22. Answer the following questions:
4.2.1) For which values of A will S = {A1, A2, A3, A4} form a linearly independent
set? Show all calculations.
a
4.2.2) Let B
be a matrix in Mm. Set up a linear system of equations that
d
need to be satisfied such that B is spanned by the vectors in S.
4.2.3) Use the system in question 4.2.2 to find out if there are values of A for which
the set S = {A1, A2, A3, A4} will be a basis for M22. If there are such values, find
them all.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d9956ec-140c-4550-9030-c03bb17d94a9%2Fc9d2fc98-5774-43c8-bb86-4c24f3225e92%2Fvs7t47f_processed.png&w=3840&q=75)
Transcribed Image Text:4) Consider the vector space M22 of all 2 x 2 matrices with normal matrix addition
and scalar multiplication as the operations. Answer the following questions:
T
T
1
4.1) Let W be the set of all matrices A E M22 such that ( A
= AT
Prove or disprove that W is a subspace of M2.
A1 =
[2 A]
A2
3 0
[o 2]
A3
4.2)
Let
and A4
be matrices in
M22. Answer the following questions:
4.2.1) For which values of A will S = {A1, A2, A3, A4} form a linearly independent
set? Show all calculations.
a
4.2.2) Let B
be a matrix in Mm. Set up a linear system of equations that
d
need to be satisfied such that B is spanned by the vectors in S.
4.2.3) Use the system in question 4.2.2 to find out if there are values of A for which
the set S = {A1, A2, A3, A4} will be a basis for M22. If there are such values, find
them all.
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