How would you prove this theorem:
The column space of an m x n matrix A is a subspace of R^m
by using this definition: A subspace of a vector space V is a subset H of V that has three properties: a) the zero vector of V is in H. b) H is closed under vector addition. c) H is closed under multiplication by scalars.
To prove the theorem that the column space of an m x n matrix A is a subspace of R^m, we need to show that it satisfies the three properties of a subspace.
1) The zero vector of R^m is in the column space of A:
The zero vector in R^m is the vector where all entries are zero. Let's denote it by 0_m. Since the column space of A consists of all possible linear combinations of the columns of A, we can construct the zero vector in the column space by multiplying every column of A by zero and adding them together:
0_m = 0 * column_1 + 0 * column_2 + ... + 0 * column_n
Since all coefficients multiplying the columns are zero, we can see that the zero vector is indeed in the column space.
2) The column space of A is closed under vector addition:
To show that the column space is closed under vector addition, we need to take any two vectors in the column space, say vector x and vector y, and show that their sum is also in the column space.
Let x and y be vectors in the column space of A, represented as x = A * c and y = A * d, where c and d are column vectors of appropriate size. We can now express their sum as:
x + y = (A * c) + (A * d)
Using distributive property of matrix multiplication over addition, we can rewrite this as:
x + y = A * (c + d)
This shows that the sum of vectors x and y is also in the column space since it can be represented as the product of matrix A and another column vector.
3) The column space of A is closed under scalar multiplication:
To demonstrate that the column space is closed under scalar multiplication, we must take any vector in the column space, say vector x, and show that multiplying it by a scalar c also results in a vector in the column space.
Let x be a vector in the column space of A, represented as x = A * c, where c is a column vector. We can now express the scalar multiple of x as:
c * x = c * (A * c)
Using the associative property of matrix multiplication, we can rearrange this as:
c * x = (c * A) * x
This shows that the scalar multiple of vector x is also in the column space since it can be represented as the product of matrix (c * A) and column vector x.
By satisfying the three properties of a subspace - containing the zero vector, closure under vector addition, and closure under scalar multiplication - we have shown that the column space of an m x n matrix A is indeed a subspace of R^m.