Question

Let A be a 2x2 matrix and W = {x=(x1,x2):Ax+0 }. Show that W is a subspace of the vector space R2

Answer 1

it that supposed to say Ax=0?

Answer 2

yes

Answer 3

Assuming it is Ax=0, you need to answer three questions:

Is the zero vector in W? does:\[A\cdot \vec{0}=\vec{0}?\]
If x1 is in W, and x2 is in W, is x1+x2 in W? in other words if:\[A\vec{x_1}=0,A\vec{x_2}=0\]does:\[A(\vec{x_1}+\vec{x_2})=0?\]
Then if c is any scalar, and x1 is in W, is cx1 in W? in other words if:\[c\in \mathbb{R},A\vec{x_1}=0\]does:\[A(c\vec{x_1})=0?\]if the answer is yes to all three of those questions, then W is a subspace.

Answer 4

Thank you. I will try to do that. Can you help me with the other one also. I just can't figure it out how to do it.

Answer 5

Consider the vector space M 2,2 with the standard operations of matrices addition and scalar multiplication. Let W = {[a,b,c,d]} : a+d=1}. Is W a subspace of M 2,2 where {[a,b,c,d]} is a (x1,x2,x3,x4) 2x2 matrix

Answer 6

The first question we need to ask is "Is the 'zero vector' in this space?" Since we are talking about a space of matrices, and not vectors in particular, we are really asking, "Is the 'zero matrix' in this space?". So look at the zero matrix, is a + d = 1 in the zero matrix?

Answer 7

so that's all I need to show? a+d doesn't =1? How do I show this? I'm sorry we just started this chapter and I'm so confused..

Answer 8

That is all you need to show. and to show it, just note that the zero matrix has 0's for all its entries. a = 0, b = 0, c = 0, and d = 0. So a+d = 0+0 = 0. Not 1. Since W is the set of all matrices such that a + d = 1, the zero matrix isnt in that set, so W couldnt possibly be subspace.

Answer 9

Thank you so much for your help :)