Subspaces

Question 1

W1 and W2 are subspaces of V. What is the intersection defined as?

Answer 1

The set of all elements that belong to both W1 and W2. Written as W1 ⋂ W2 ={ x ∈ V | x ∈ W1 and x ∈ W2 }

Question 2

W1 and W2 are subspaces of V. What is the union defined as?

Answer 2

The set of elements that belong to atleast W1 or W2 (could belong to both)

W1 ∪ W2 = { x ∈ V | x ∈ W1 or x ∈ W2 }

Question 3

Is the intersection of W1 and W2 a subspace of V? Is the union?

Answer 3

The intersection set is but the union set is not

Question 4

What is the nullspace of a matrix?

Answer 4

If you were to treat that matrix as the standard matrix of a transformation, the kernel of that transformation would be Null(A). Essentially, it’s the set of vectors x, in R^n that make Ax=0 vector of dimension m

Question 5

What is the column space of a matrix?

Answer 5

If you were to make the matrix the standard matrix of a transformation, the range of that transformation would be Col(A). It’s the set of vectors in R^m that are a linear combo of the columns of the matrix. In otherwords, x must satisfy Ax=b for all vectors b in R^m

Question 6

Is Nul(A) a subspace of any dimension? What about Col(A)?

Answer 6

Col(A) is a subpace of R^m and Nul(A) is a subspace of R^n assuming A is a m×n matrix

Question 7

What is Row(A) and what is it a subspace of?

Answer 7

Col(A^T), R^n assuming A is m×n

Question 8

Do elementary row operations preserve the coloumn space? What about the null space?

Answer 8

The null space is preserved. If A and B are row equivalent, then Nul(A)=Nul(B). The column space does change though with row operations. Col(A) ≠ Col(B)