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

Solutions

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 1 year ago

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