Question

1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all lower triangular 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, c, 2a + 3b – 3c) (which is a subspace of R4).

Answer 1

Som we have to find the dimension and a basis for the given vector spaces. (a) Given weedor space is of all lower triangular 2x2 matrices (which is a subspace of M2x2) Let w be the vector-space of all lower triangular 2x2 matrices. then, W = oད། O GM2*2 a21 922) A= Lower-triangular $ ali Q21, C122 GR , F = field them, WE 90(10)***033*** } { [].[:$], [] Basis of w dimension of w=3 (b) Given, vector Share of all rectors of the form (a, b, c, 2a+36- bb-3c) (which is a subspace of i let, a, b, c, 29+36-30) ERY 14, ,C EF , where f-field W - wa Efa(1,0,0,.2) -(1,0, 0, 2) +b(0,1,0, 3)+c(0,0,1 Now, we check that (1,0,0,2), (0,1,0,3) and (0,011,-3) are Lineartly independent or not. So, 1 0 0 2 o 3 01 Ooi3 Pand lou 2 0103 001-3 - 3-) wrectors are linearlly- Independent 1 Since, w contains three vectors, which all are lineartly- Independent, So, Basis of w= 10,012), (0,1,0,1),(0,0,1,3 - dimension of w=3