Linear algebra Subspace help Please

This is the problem:
http://s2.postimage.org/ml6ncq4gy/smallh…

Any help would be great.
Thank you so much

Let A_1,...,A_n represent the columns of A, each of which are in R^m. Clearly col A, which is the span{A_1,...,A_n}, is a subset of R^m.

Now we need to show that col A is a subspace, not just a subset, of R^m. To do this, we need to show that col A contains zero, and is closed under addition and scalar multiplication.

By definition,

col A = span{A_1,... ,A_n}
= {c_1*A_1+...+c_n*A_n | c_1,...,c_n are real numbers}

Clearly col A contains 0, since c_1*A_1+...+c_n*A_n = 0 if c_1,...,c_n are all zero.

Col A is closed under addition because for any real numbers c_1,...,c_n and d_1,...,d_n, we have

(c_1*A_1+...+c_n*A_n)+(d_1*A_1+...+d_n…
= (c_1+d_1)*A_1+...+(c_n+d_n)*A_n, which is also in col A.

Col A is closed under scalar multiplication because for any real numbers c_1,...,c_n and r we have

r(c_1*A_1+...+c_n*A_n)
= (rc_1)*A_1+...+(rc_n)*A_n, which is also in col A.

So col A is a subspace of R^m.