Question

In: Math

Let M 2,2 be the set of all 2x2 matrices determine whether the following subspaces. A)...

Let M 2,2 be the set of all 2x2 matrices determine whether the following subspaces.

A) the set of all 2x2 diagnol matrices

B) the set of all matrices with a12 entry

C) the set of all 2x2 triangular matrices

Expert Solution

Let M _2,2 be the set of all 2x2 matrices.

A). Let V be the set of all 2x2 diagonal matrices. Let A =

a	0
0	b

and B =

c	0
0	d

be 2 arbitrary 2x2 diagonal matrices in V and let k be an arbitrary scalar. Then A+B =

a+c	0
0	b+d

which is a 2x2 diagonal matrix. It implies that A+B is in V so that V is closed under vector addition. Also, kA =

ka	0
0	kb

which is a 2x2 diagonal matrix. It implies that kA is in V so that V is closed under scalar multiplication. Further, the 2x2 zero matrix, being a diagonal matrix, is in V. Hence V is a vector space and being a subset of M _2,2 , it is a subspace of M _2,2.

B) the set of all matrices with a₁₂ entry – It is unclear. What is the a₁₂ entry ?

C) the set of all 2x2 triangular matrices

Let V be the set of all 2x2 triangular matrices.

A triangular matrix is a square matrix which is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero. Since the sum of a 2x2 lower triangular matrix and a 2x2 upper triangular matrix is not a 2x2 triangular matrix, hence V is not closed under vector addition. Hence, V is not a vector space and, therefore, not a subspace of M _2,2.

milcah answered 2 years ago

Determine which subsets are subspaces of M 2x2 (R) and prove your answer. a. W =...

Determine which subsets are subspaces of M 2x2 (R) and prove your answer. a. W = {A ∈ M 2x2 (R) | a12 = -a21} b. W = {A ∈ M 2X2 (R) | a12 = 1} c. Fix B ∈ M 2x2 (R). Let W ={ A ∈ M 2x2 (R) | AB = BA

Determine if the following subsets are subspaces: 1. The set of grade 7 polynomials 2. The...

Determine if the following subsets are subspaces: 1. The set of grade 7 polynomials 2. The set of polynomials of degree 5 such that P (0) = 0 3. The set of continuous functions such that f (0) = 2

8. Let G = GL2(R). (Tables are actually matrices) (d) Determine whether A = { a...

8. Let G = GL2(R). (Tables are actually matrices) (d) Determine whether A = { a b 0 0 } | a, b in R ) is a subgroup of G. (e) Determine whether B = { 0 b c 0 } | b, c in R ) is a subgroup of G. (f) Determine whether C = { 1 0 0 d } | d in R, d not equal 0) is a subgroup of G.

Determine if the vector(s), polynomial(s), matrices are linearly independent in R^3, P3(R), M 2x2 (R). Show...

Determine if the vector(s), polynomial(s), matrices are linearly independent in R^3, P3(R), M 2x2 (R). Show algebraically how you found your answer. a. < 2, 1, 5 > , < -2, 3, 1 > , < -4, 4, 1 > b. x^3 - 3x^2 + 2x +1, -2x^3 + 9x^2 -3, x^3 + 6x c. | 1 2 | | -3, -1 | | -4 2 | , | 2 1|

Let E be the set of all positive integers. Define m to be an "even prime"...

Let E be the set of all positive integers. Define m to be an "even prime" if m is even but not factorable into two even numbers. Prove that some elements of E are not uniquely representable as products of "even primes." Please be as detailed as possible!

Find the set of ALL optimal solutions to the following LP: min z= 3x1−2x2 subject to...

Find the set of ALL optimal solutions to the following LP: min z= 3x1−2x2 subject to 3x1+x2≤12 3x1−2x2−x3= 12 x1≥2 x1, x2, x3≥0

Let x ∈ Rn be any nonzero vector. Let W ⊂ Rnxn consist of all matrices...

Let x ∈ Rn be any nonzero vector. Let W ⊂ Rnxn consist of all matrices A such that Ax = 0. Show that W is a subspace and find its dimension.

Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or...

Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if a) a is taller than b. b) a and b were born on the same day.

Given two 2x2 matrices: M=1 21 1,N=-1 2 1 -1 (a).Verify multiplication of M x N...

Given two 2x2 matrices: M=1 21 1,N=-1 2 1 -1 (a).Verify multiplication of M x N mod 26 = N x M mod 26 = I (identity matrix) (b). Use Hill cipher to encrypt the message EXAMS

Let S be a set of n numbers. Let X be the set of all subsets...

Let S be a set of n numbers. Let X be the set of all subsets of S of size k, and let Y be the set of all ordered k-tuples (s1, s2, , sk) such that s1 < s2 < < sk. That is, X = {{s1, s2, , sk} | si S and all si's are distinct}, and Y = {(s1, s2, , sk) | si S and s1 < s2 < < sk}. (a) Define a one-to-one correspondence f : X → Y. Explain...