2. Suppose that A is a m x n matrix. If x is a vector in IRT, then Ax will be a

Question

2. Suppose that A is a m x n matrix. If x is a vector in IRT, then Ax will be a vector in R. Define the range of A to be the set of all vectors in R"m that can be obtained in this way. In set notation: range(A) = {w E R"( w = Ax for some x ER") 3. 13 points Use the subspace test to explain why range(4) is a subspace of R". 3 points Now suppose that m = n (so A is square), and also assume A is invertible. Carefully explain why range(A)-R". 4. (Because you already know that range(A) is always contained in " (in symbols, range(A) C R) prove that these two sets are equal, you only need to explain why every vector in R" is also in range(A) In other words, you need to show that is a subset of range(A) ( C range(A))

Explanation / Answer

Let, u,v be in Range(A) and c be a real number

So there exist, x,y so that

u=Ax,v=Ay

u+v=Ax+Ay=A(x+y)

Hence, u+v is in range(A)

cu=cAx=A(cx)

Hence, cu is in range(A)

Hence, range(A) is a subspace

4.

By rank nullity theorem

dim range(A)+dim null (A)=n=m

But since A is invertible so null(A)={0}

Hence, dim null (A)=0

Hence, dim range(A)=m=n

Hence, range(A)=Rm

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