- #1

- 489

- 0

Consider 2-by-2 matrices [itex]\mathbf{A} =\left( \begin{array}{cc}a & b \\c & d \\\end{array} \right) \in \mathbbm{R}^{2 X 2}[/itex]. Which of the following are subspaces of [itex]\mathbbm{R}^{2 X 2}[/itex]?

(A) {A | c = 0}

(B) {A | a + d = 0}

(C) {A | ad - bc = 0}

(D) {A | b = c}

(E) {A | Av = 2v}, where v is some vector in [itex]R^2[/itex].

I think that all except the last would be subspaces of the 2 X 2 set of matrices. I know that choice (C) is the determinent of the matrix, which is in the subspace of it's own matrix, right?

Also, how would I make a case for the first four choices using the 3 "rules" that determine if a vector is a subspace of another?

(1) {0} must be in the subset for it to be a subset.

(2) if v is in W then a * v is in W for all real numbers a.

(3) if u and v are in W, then u + v is in W.

(A) {A | c = 0}

(B) {A | a + d = 0}

(C) {A | ad - bc = 0}

(D) {A | b = c}

(E) {A | Av = 2v}, where v is some vector in [itex]R^2[/itex].

I think that all except the last would be subspaces of the 2 X 2 set of matrices. I know that choice (C) is the determinent of the matrix, which is in the subspace of it's own matrix, right?

Also, how would I make a case for the first four choices using the 3 "rules" that determine if a vector is a subspace of another?

(1) {0} must be in the subset for it to be a subset.

(2) if v is in W then a * v is in W for all real numbers a.

(3) if u and v are in W, then u + v is in W.

Last edited: